1 00:00:00,000 --> 00:00:00,030 2 00:00:00,030 --> 00:00:02,330 The following content is provided under a Creative 3 00:00:02,330 --> 00:00:03,710 Commons license. 4 00:00:03,710 --> 00:00:05,450 Your support will help MIT OpenCourseWare 5 00:00:05,450 --> 00:00:09,395 continue to offer high quality educational resources for free. 6 00:00:09,395 --> 00:00:11,520 To make a donation, or to view additional materials 7 00:00:11,520 --> 00:00:15,246 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,246 --> 00:00:15,870 at ocw.mit.edu. 9 00:00:15,870 --> 00:00:20,880 10 00:00:20,880 --> 00:00:25,210 PROFESSOR STRANG: So I'm ready for anything, hope I am. 11 00:00:25,210 --> 00:00:27,030 Questions about any topic. 12 00:00:27,030 --> 00:00:27,530 Yes. 13 00:00:27,530 --> 00:00:28,816 AUDIENCE: [INAUDIBLE] 14 00:00:28,816 --> 00:00:31,190 PROFESSOR STRANG: I feel this is like a White House press 15 00:00:31,190 --> 00:00:31,730 conference. 16 00:00:31,730 --> 00:00:35,760 I think there's always somebody in the front row who 17 00:00:35,760 --> 00:00:38,610 gets to ask the first question, and then 18 00:00:38,610 --> 00:00:41,350 gets to say thank you Mr. President at the end, 19 00:00:41,350 --> 00:00:47,450 and then I'm off. 20 00:00:47,450 --> 00:00:50,800 Yes. 21 00:00:50,800 --> 00:00:53,100 I'm tempted by the way to ask you all, 22 00:00:53,100 --> 00:00:55,060 are you going to vote next Tuesday 23 00:00:55,060 --> 00:00:57,780 and of course I'd like to know who you vote for, 24 00:00:57,780 --> 00:01:00,130 and I'd like to give you my advice. 25 00:01:00,130 --> 00:01:04,050 But I don't know that that's proper. 26 00:01:04,050 --> 00:01:06,510 If anybody wants advice, they can email. 27 00:01:06,510 --> 00:01:08,460 But please vote. 28 00:01:08,460 --> 00:01:09,320 Please vote. 29 00:01:09,320 --> 00:01:10,380 Yeah. 30 00:01:10,380 --> 00:01:13,900 Alright, question here and then we'll have, well, yeah. 31 00:01:13,900 --> 00:01:19,360 AUDIENCE: [INAUDIBLE] 32 00:01:19,360 --> 00:01:21,660 PROFESSOR STRANG: Oh, OK, so those were just posted. 33 00:01:21,660 --> 00:01:24,930 Like, I see. 34 00:01:24,930 --> 00:01:29,400 3.3 number two. 35 00:01:29,400 --> 00:01:33,970 AUDIENCE: [INAUDIBLE] 36 00:01:33,970 --> 00:01:40,130 PROFESSOR STRANG: OK, so this was a case -- yeah, right. 37 00:01:40,130 --> 00:01:42,640 Oh, OK. 38 00:01:42,640 --> 00:01:50,690 We could be wrong so this is 3.3 number two, 39 00:01:50,690 --> 00:01:55,970 asks you about the flow field. 40 00:01:55,970 --> 00:02:00,540 Which has no flow in the x direction, 41 00:02:00,540 --> 00:02:03,530 the velocity in the x direction is zero. 42 00:02:03,530 --> 00:02:06,660 The velocity in the y direction is x. 43 00:02:06,660 --> 00:02:10,210 OK, so suppose we just take that as a full field 44 00:02:10,210 --> 00:02:13,820 and try to understand, is it a gradient? 45 00:02:13,820 --> 00:02:16,260 I mean, so what are the questions I would ask? 46 00:02:16,260 --> 00:02:17,940 Is it a gradient of anything? 47 00:02:17,940 --> 00:02:23,290 Because we're now thinking v and w are pretty much the same guy. 48 00:02:23,290 --> 00:02:26,060 So is it a gradient, yes or no? 49 00:02:26,060 --> 00:02:29,110 If so, what's the potential? 50 00:02:29,110 --> 00:02:32,160 Is it divergence-free, yes or no? 51 00:02:32,160 --> 00:02:34,810 If it is, what's the stream function? 52 00:02:34,810 --> 00:02:40,510 And of course if the answer to both tests was yes, then 53 00:02:40,510 --> 00:02:44,110 we would be talking about Laplace's equation. 54 00:02:44,110 --> 00:02:49,600 I suspect for this example the answer, 55 00:02:49,600 --> 00:02:52,480 at least to one of the two questions, will be no. 56 00:02:52,480 --> 00:02:58,160 So we won't have the two pieces coming together into Laplace. 57 00:02:58,160 --> 00:02:59,820 OK, so first of all. 58 00:02:59,820 --> 00:03:01,860 Is it a gradient? 59 00:03:01,860 --> 00:03:08,300 What's the test for, so my two questions are, 60 00:03:08,300 --> 00:03:13,160 is v the gradient of some u? 61 00:03:13,160 --> 00:03:17,560 And what's the test for that? 62 00:03:17,560 --> 00:03:19,930 You remember if it is a gradient -- 63 00:03:19,930 --> 00:03:21,860 and see if I can remember myself. 64 00:03:21,860 --> 00:03:28,130 If it is a gradient, then this is du/dx, and this is du/dy, 65 00:03:28,130 --> 00:03:33,050 and the condition that v_1 and v_2 would have to satisfy 66 00:03:33,050 --> 00:03:36,090 is that the y derivative of that would 67 00:03:36,090 --> 00:03:38,820 have to equal the x derivative of that, 68 00:03:38,820 --> 00:03:42,840 because on the right-hand side they are the same. 69 00:03:42,840 --> 00:03:45,550 u_xy is the same as u_yx. 70 00:03:45,550 --> 00:03:53,300 So I would look at -- so let me write that again. 71 00:03:53,300 --> 00:04:04,620 I need dv_1/dy to equal dv_2/dx, and is that true in this 72 00:04:04,620 --> 00:04:06,690 example? 73 00:04:06,690 --> 00:04:09,780 What's dv_1/dy? 74 00:04:09,780 --> 00:04:10,546 Zero. 75 00:04:10,546 --> 00:04:11,170 What's dv_2/dx? 76 00:04:11,170 --> 00:04:14,610 77 00:04:14,610 --> 00:04:15,600 One. 78 00:04:15,600 --> 00:04:17,960 So the answer's no. 79 00:04:17,960 --> 00:04:19,360 OK. 80 00:04:19,360 --> 00:04:22,870 So, test failed. 81 00:04:22,870 --> 00:04:29,480 Alright, the second question is does it possibly sit over 82 00:04:29,480 --> 00:04:31,680 in the divergence-free world? 83 00:04:31,680 --> 00:04:36,310 Is the divergence of, now I'll call it w. equal zero? 84 00:04:36,310 --> 00:04:39,420 So the answer was no to that question 85 00:04:39,420 --> 00:04:43,720 but now I think the answer to this question will be yes. 86 00:04:43,720 --> 00:04:47,550 Because what's the divergence of this thing? 87 00:04:47,550 --> 00:04:51,020 It's the x derivative of that, which is certainly 88 00:04:51,020 --> 00:04:55,780 zero, plus the y derivative of that, which is certainly zero. 89 00:04:55,780 --> 00:04:58,260 So the answer is yes. 90 00:04:58,260 --> 00:05:04,290 So there's no potential but there is a stream function, 91 00:05:04,290 --> 00:05:04,980 right? 92 00:05:04,980 --> 00:05:08,200 Because the stream function comes in with this test. 93 00:05:08,200 --> 00:05:12,390 So let's remember what, just from today's lecture, what 94 00:05:12,390 --> 00:05:13,980 was the stream function? 95 00:05:13,980 --> 00:05:26,550 From dw_1/dx+dw_2/dy=0, that'll be satisfied if w_1 is the y 96 00:05:26,550 --> 00:05:28,160 derivative of the stream function, 97 00:05:28,160 --> 00:05:32,220 and w_2 is minus the x derivative. 98 00:05:32,220 --> 00:05:38,730 Because then this matches the x derivative of this 99 00:05:38,730 --> 00:05:41,980 plus the y derivative of this, which is the divergence; 100 00:05:41,980 --> 00:05:44,980 on the right-hand side I would get zero. 101 00:05:44,980 --> 00:05:47,040 So there's got to be an s, and what is it? 102 00:05:47,040 --> 00:05:49,150 Probably not hard to find. 103 00:05:49,150 --> 00:05:50,930 Let's see. 104 00:05:50,930 --> 00:05:53,540 Here w_1 is zero, so that tells me 105 00:05:53,540 --> 00:05:57,600 s doesn't depend on y at all. w_2 106 00:05:57,600 --> 00:06:02,860 is x, so x is supposed to be minus the x derivative 107 00:06:02,860 --> 00:06:05,930 of the stream function, so what is the stream function now? 108 00:06:05,930 --> 00:06:10,330 Have I got room to put it here? 109 00:06:10,330 --> 00:06:11,820 Just about. 110 00:06:11,820 --> 00:06:16,440 What will work? 111 00:06:16,440 --> 00:06:21,060 So again, here's w_1. 112 00:06:21,060 --> 00:06:25,420 The y derivative of s is zero. w_2 tells me 113 00:06:25,420 --> 00:06:28,710 that the x derivative of s is minus x. 114 00:06:28,710 --> 00:06:31,660 So all I'm looking for is a function that only depends 115 00:06:31,660 --> 00:06:36,520 on x, has no dependence on y, and its derivative 116 00:06:36,520 --> 00:06:38,530 should be minus x. 117 00:06:38,530 --> 00:06:41,260 So what's the function? 118 00:06:41,260 --> 00:06:45,070 Minus a half of x squared. 119 00:06:45,070 --> 00:06:54,430 Yeah, so this gives me s equal minus a half of x squared. 120 00:06:54,430 --> 00:06:58,830 Alright, so you're saying that, so there 121 00:06:58,830 --> 00:07:02,560 is a stream function, right. 122 00:07:02,560 --> 00:07:04,780 And what does that travel along? 123 00:07:04,780 --> 00:07:09,110 That travels along steam, that means that the flow buzzes 124 00:07:09,110 --> 00:07:10,800 along streamlines. 125 00:07:10,800 --> 00:07:12,650 And what are the streamlines? 126 00:07:12,650 --> 00:07:16,220 They're the lines where s is constant. 127 00:07:16,220 --> 00:07:19,100 Equipotentials were the lines where u is constant, 128 00:07:19,100 --> 00:07:22,050 but here we don't have a u in this problem. 129 00:07:22,050 --> 00:07:25,300 Streamlines are lines where the s is constant, 130 00:07:25,300 --> 00:07:28,030 so minus 1/2 x squared is constant, 131 00:07:28,030 --> 00:07:31,270 what's the picture look like? 132 00:07:31,270 --> 00:07:35,910 Picture then, for that, well, and you 133 00:07:35,910 --> 00:07:42,600 know what the flow is doing at a typical point here. 134 00:07:42,600 --> 00:07:46,390 Say x=3, y=1. 135 00:07:46,390 --> 00:07:49,630 Let me draw the little arrow. 136 00:07:49,630 --> 00:07:55,150 With a big chalk Which way is the flow going? 137 00:07:55,150 --> 00:08:01,940 Well, the x component is zero, the y component is x. 138 00:08:01,940 --> 00:08:05,730 So the flow is going up there, right? 139 00:08:05,730 --> 00:08:10,270 Here the y component's x. 140 00:08:10,270 --> 00:08:16,330 This whole line is all traveling up with the same velocity. 141 00:08:16,330 --> 00:08:19,770 If I drop a leaf there, it buzzes up that straight line. 142 00:08:19,770 --> 00:08:21,570 So that's the streamline. 143 00:08:21,570 --> 00:08:27,940 And its velocity is, that's x equal -- if, say, 144 00:08:27,940 --> 00:08:33,590 the velocity is three then this speed is three. 145 00:08:33,590 --> 00:08:35,440 It's going up that line. 146 00:08:35,440 --> 00:08:38,160 So that's a streamline. 147 00:08:38,160 --> 00:08:41,780 And sure enough, on that line minus 1/2 x squared 148 00:08:41,780 --> 00:08:42,950 is a constant. 149 00:08:42,950 --> 00:08:45,850 So you see we're not talking parabolas 150 00:08:45,850 --> 00:08:51,200 here because our curve is not y equals minus 1/2 x squared, 151 00:08:51,200 --> 00:08:54,140 it's minus 1/2 x squared equal constant. 152 00:08:54,140 --> 00:08:55,330 Yeah, that's what we want. 153 00:08:55,330 --> 00:09:02,630 So, but now having got so far, let me take x=1, say. 154 00:09:02,630 --> 00:09:04,650 What's the flow like on that? 155 00:09:04,650 --> 00:09:11,340 So there's a streamline with s equal constant. 156 00:09:11,340 --> 00:09:14,070 And the velocity on that is zero, 157 00:09:14,070 --> 00:09:17,750 so nothing is going in that -- horizontally. 158 00:09:17,750 --> 00:09:25,120 And now it's one, so the flow is slower up this line. 159 00:09:25,120 --> 00:09:26,640 OK, slower flow. 160 00:09:26,640 --> 00:09:30,140 This was faster flow. 161 00:09:30,140 --> 00:09:34,610 And then the question that's in that homework problem is, 162 00:09:34,610 --> 00:09:38,570 is there any rotation in this flow? 163 00:09:38,570 --> 00:09:45,300 We think about rotation, we have an image of rotational flow. 164 00:09:45,300 --> 00:09:49,370 And that could be the next example, we could figure out. 165 00:09:49,370 --> 00:09:54,660 A flow that goes around in circles, right? 166 00:09:54,660 --> 00:09:56,440 Those would be the streamlines. 167 00:09:56,440 --> 00:10:03,830 So this would be like pure rotation, shall I call it. 168 00:10:03,830 --> 00:10:05,330 But I don't have that there. 169 00:10:05,330 --> 00:10:08,580 I just want to draw the other picture, in which 170 00:10:08,580 --> 00:10:16,410 the streamlines are circles. 171 00:10:16,410 --> 00:10:19,790 To have another nice, clean, beautiful example. 172 00:10:19,790 --> 00:10:21,910 OK, but here we don't have. 173 00:10:21,910 --> 00:10:24,070 Our streamlines are straight lines, 174 00:10:24,070 --> 00:10:26,910 and yet we have rotation. 175 00:10:26,910 --> 00:10:30,490 That's the point here. 176 00:10:30,490 --> 00:10:32,180 Why do I say we have rotation? 177 00:10:32,180 --> 00:10:36,530 Because the test for rotation was that original test 178 00:10:36,530 --> 00:10:43,710 of looking at, which I just wrote the answer to be no here. 179 00:10:43,710 --> 00:10:47,120 So if it's not a gradient, the reason 180 00:10:47,120 --> 00:10:48,530 is there's some rotation. 181 00:10:48,530 --> 00:10:51,230 Gradient fields don't have any rotation. 182 00:10:51,230 --> 00:10:59,680 The rotation is this thing that comes out, 183 00:10:59,680 --> 00:11:04,000 yeah it's this difference. dv_2/dx, 184 00:11:04,000 --> 00:11:06,720 it's the difference between those that tells us 185 00:11:06,720 --> 00:11:08,190 the rotation. 186 00:11:08,190 --> 00:11:12,260 And that was not zero, right? 187 00:11:12,260 --> 00:11:18,600 For this example dv_1/dy was zero, because v_1 is zero. 188 00:11:18,600 --> 00:11:23,670 dv_2/dx was one, because v_2 is x. 189 00:11:23,670 --> 00:11:25,590 So there's some rotation here. 190 00:11:25,590 --> 00:11:30,100 And in other words the test for being a gradient 191 00:11:30,100 --> 00:11:31,800 is no rotation. 192 00:11:31,800 --> 00:11:33,720 This fails that test. 193 00:11:33,720 --> 00:11:35,260 But how is it rotating? 194 00:11:35,260 --> 00:11:41,000 How can it be rotating when the all the flow is just 195 00:11:41,000 --> 00:11:43,530 traveling vertically? 196 00:11:43,530 --> 00:11:48,150 I guess I give you this example because it meant something 197 00:11:48,150 --> 00:11:49,820 to me. 198 00:11:49,820 --> 00:11:54,350 My image of rotation was this simpleminded type of flow. 199 00:11:54,350 --> 00:11:56,940 You know, like a phonograph record or something. 200 00:11:56,940 --> 00:11:59,160 This would be called a sheer flow. 201 00:11:59,160 --> 00:12:03,950 A very important type of flow. 202 00:12:03,950 --> 00:12:08,540 And actually, you'll realize that if x is negative 203 00:12:08,540 --> 00:12:13,480 then the flow in the second component, the velocity, 204 00:12:13,480 --> 00:12:14,470 is now negative. 205 00:12:14,470 --> 00:12:17,300 So it would be the streamline, the flow 206 00:12:17,300 --> 00:12:19,820 would be going down this way. 207 00:12:19,820 --> 00:12:22,420 And this point wouldn't move at all. 208 00:12:22,420 --> 00:12:24,670 This would be, well I don't know if it's a streamline, 209 00:12:24,670 --> 00:12:30,200 it's a stagnant streamline, right? x=0. 210 00:12:30,200 --> 00:12:33,510 On that line, there's no velocity. [0, 0]. 211 00:12:33,510 --> 00:12:37,000 So this is all just staying there. 212 00:12:37,000 --> 00:12:40,000 These lines are moving, this line moving faster, 213 00:12:40,000 --> 00:12:41,950 this line would be moving even faster. 214 00:12:41,950 --> 00:12:44,920 This line's going the other way. 215 00:12:44,920 --> 00:12:46,650 Faster and faster the other way. 216 00:12:46,650 --> 00:12:49,190 It's a important flow. 217 00:12:49,190 --> 00:12:52,650 You know, in earthquakes and things like that. 218 00:12:52,650 --> 00:12:57,920 This happens, when one plate shears with respect to another. 219 00:12:57,920 --> 00:12:59,210 So that's shearing. 220 00:12:59,210 --> 00:13:05,940 The word shearing means that a line that was -- 221 00:13:05,940 --> 00:13:09,810 that line after a while is tilted. 222 00:13:09,810 --> 00:13:12,050 This is going faster than this. 223 00:13:12,050 --> 00:13:12,710 Yes. 224 00:13:12,710 --> 00:13:16,440 AUDIENCE: [INAUDIBLE] 225 00:13:16,440 --> 00:13:20,530 PROFESSOR STRANG: Right, being a constant. 226 00:13:20,530 --> 00:13:24,690 AUDIENCE: [INAUDIBLE] That's true. 227 00:13:24,690 --> 00:13:28,370 Ah, well, OK. 228 00:13:28,370 --> 00:13:32,290 Let's see. 229 00:13:32,290 --> 00:13:33,590 Well, how do I fix that? 230 00:13:33,590 --> 00:13:39,770 AUDIENCE: [INAUDIBLE] 231 00:13:39,770 --> 00:13:41,810 PROFESSOR STRANG: Yes. 232 00:13:41,810 --> 00:13:42,870 That's a good question. 233 00:13:42,870 --> 00:13:48,680 Should I have allowed in my stream function, 234 00:13:48,680 --> 00:13:51,320 I mean that's a stream function. 235 00:13:51,320 --> 00:13:54,730 Because it satisfies the equations that stream functions 236 00:13:54,730 --> 00:13:57,580 are -- I could have thrown in a constant, yeah. 237 00:13:57,580 --> 00:13:59,910 So your pointing out a difficulty 238 00:13:59,910 --> 00:14:02,880 makes me think I should have thrown in a constant. 239 00:14:02,880 --> 00:14:07,510 So if I throw in constants then I could get other lines, yeah. 240 00:14:07,510 --> 00:14:09,630 Thanks, that's a good point. 241 00:14:09,630 --> 00:14:13,600 I just want to see, do you see rotation in this flow, 242 00:14:13,600 --> 00:14:15,380 in this shear flow? 243 00:14:15,380 --> 00:14:16,600 And I think you do. 244 00:14:16,600 --> 00:14:17,840 If you think about it. 245 00:14:17,840 --> 00:14:22,680 Suppose you put a little leaf, or a little penny or something 246 00:14:22,680 --> 00:14:23,920 right there. 247 00:14:23,920 --> 00:14:27,150 OK, is it going to turn? 248 00:14:27,150 --> 00:14:31,880 It'll flow along, but as it flows, is it going to turn? 249 00:14:31,880 --> 00:14:40,410 In other words, is there some difference in the speed on one 250 00:14:40,410 --> 00:14:41,550 side compared to the other? 251 00:14:41,550 --> 00:14:45,280 I mean, it's what makes a curveball curve, right? 252 00:14:45,280 --> 00:14:50,140 When the pitcher throws the ball, he imparts a spin to it, 253 00:14:50,140 --> 00:14:52,090 and that gives a different pressure 254 00:14:52,090 --> 00:14:56,630 on the two sides of the ball, and the ball moves. 255 00:14:56,630 --> 00:14:59,330 I think that's going to happen here. 256 00:14:59,330 --> 00:15:02,540 Maybe you see it, and I'm just talking. 257 00:15:02,540 --> 00:15:05,220 I mean, this side is going faster than this side. 258 00:15:05,220 --> 00:15:11,180 So the net result is that even though the thing is traveling 259 00:15:11,180 --> 00:15:15,440 up and down, it's turning. 260 00:15:15,440 --> 00:15:18,550 It's turning because the right-hand side is going faster 261 00:15:18,550 --> 00:15:19,910 than the left-hand side. 262 00:15:19,910 --> 00:15:22,750 So it does have a rotation. 263 00:15:22,750 --> 00:15:28,290 This quantity, this difference between dv_1/dy and dv_2/dx, 264 00:15:28,290 --> 00:15:31,510 which is the component of the curl, 265 00:15:31,510 --> 00:15:33,700 maybe the sign should be the opposite, 266 00:15:33,700 --> 00:15:37,650 maybe I think it should be minus this plus this or something. 267 00:15:37,650 --> 00:15:40,990 Point is that it's not zero. 268 00:15:40,990 --> 00:15:43,800 So there is curl, there is rotation. 269 00:15:43,800 --> 00:15:44,830 OK. 270 00:15:44,830 --> 00:15:48,600 I was going to ask about this picture, too. 271 00:15:48,600 --> 00:15:50,870 And then I'll open to more examples. 272 00:15:50,870 --> 00:15:53,600 I just feel examples are good. 273 00:15:53,600 --> 00:15:57,430 Simple velocity fields, like 0x. 274 00:15:57,430 --> 00:16:00,040 275 00:16:00,040 --> 00:16:04,440 Just to think through, OK, what does that mean? 276 00:16:04,440 --> 00:16:06,310 Is it curl free? 277 00:16:06,310 --> 00:16:09,650 Another way of saying is it a gradient field would be to say 278 00:16:09,650 --> 00:16:11,900 is it curl free? 279 00:16:11,900 --> 00:16:15,260 Irrotational is the right word here. 280 00:16:15,260 --> 00:16:18,960 Test one, is it irrotational, answer no. 281 00:16:18,960 --> 00:16:22,320 Is it divergence-free, is it source-free, 282 00:16:22,320 --> 00:16:25,690 the answer was yes, for this example. 283 00:16:25,690 --> 00:16:28,830 If we pick another example I could reverse those, 284 00:16:28,830 --> 00:16:31,810 or another example -- I can probably come up with 285 00:16:31,810 --> 00:16:33,590 an example here. 286 00:16:33,590 --> 00:16:39,160 Let's see, what if I wanted the streamlines to be circles, 287 00:16:39,160 --> 00:16:46,110 what would be a good velocity field that goes in circles? 288 00:16:46,110 --> 00:16:47,020 Let's see. 289 00:16:47,020 --> 00:16:52,120 At a typical point, if I want the velocity to be going that 290 00:16:52,120 --> 00:16:58,030 way, here's the vector, the position vector, 291 00:16:58,030 --> 00:17:01,260 the radial vector that goes -- so what are the components 292 00:17:01,260 --> 00:17:02,880 of this vector? 293 00:17:02,880 --> 00:17:04,890 Just [x, y]. 294 00:17:04,890 --> 00:17:12,300 So now if I want the velocity field to go other way, 295 00:17:12,300 --> 00:17:18,560 what would be a good thing with rotation? 296 00:17:18,560 --> 00:17:20,640 [-y, x] would sound good. 297 00:17:20,640 --> 00:17:21,460 v=[-y, x]. 298 00:17:21,460 --> 00:17:24,440 299 00:17:24,440 --> 00:17:29,390 So are we expecting this to be a gradient of anything? 300 00:17:29,390 --> 00:17:32,730 I'm not. 301 00:17:32,730 --> 00:17:35,380 We've built in rotation here. 302 00:17:35,380 --> 00:17:38,940 I'm expecting the curl of this thing, this quantity, 303 00:17:38,940 --> 00:17:44,720 I think I take the x derivative -- 304 00:17:44,720 --> 00:17:48,805 I look at the y derivative of this and compare it with the x 305 00:17:48,805 --> 00:17:49,910 derivative of that. 306 00:17:49,910 --> 00:17:52,450 And they're not the same; in fact one is minus one 307 00:17:52,450 --> 00:17:54,070 and the other's plus one. 308 00:17:54,070 --> 00:17:56,420 So I've got rotation here. 309 00:17:56,420 --> 00:18:03,340 I've got sort of two, is the component of the curl. 310 00:18:03,340 --> 00:18:09,180 So let's just write it down. dv_2/dx, 311 00:18:09,180 --> 00:18:16,210 this vorticity that measures the turning speed is one from 312 00:18:16,210 --> 00:18:20,620 dv_2/dx, minus one is two. 313 00:18:20,620 --> 00:18:23,940 So it's not a gradient of anything. 314 00:18:23,940 --> 00:18:27,180 If the x derivative of u is minus y, 315 00:18:27,180 --> 00:18:30,890 then the y derivative can't be plus x, no good. 316 00:18:30,890 --> 00:18:35,580 OK, what about, is it divergence-free? 317 00:18:35,580 --> 00:18:38,330 318 00:18:38,330 --> 00:18:41,350 Do I need a source to keep this flow going? 319 00:18:41,350 --> 00:18:42,700 Well, what's the test? 320 00:18:42,700 --> 00:18:47,650 In other words, is there a stream function for this guy? 321 00:18:47,650 --> 00:18:50,320 I think probably there is. 322 00:18:50,320 --> 00:18:54,900 What's the test to know if there is a stream function? 323 00:18:54,900 --> 00:18:58,100 I take the divergence, I'm over on the right-hand side 324 00:18:58,100 --> 00:18:59,260 of my picture now. 325 00:18:59,260 --> 00:19:01,170 I take the divergence. 326 00:19:01,170 --> 00:19:07,130 Divergence of this v is the x derivative of that 327 00:19:07,130 --> 00:19:11,280 plus the y derivative of that, good, zero. 328 00:19:11,280 --> 00:19:13,440 So there is a stream function. 329 00:19:13,440 --> 00:19:15,620 And what is it? 330 00:19:15,620 --> 00:19:20,740 Well, I'm pretty sure that these streamlines are circles, 331 00:19:20,740 --> 00:19:22,525 I think the stream function is going 332 00:19:22,525 --> 00:19:27,670 to be x squared plus y squared. 333 00:19:27,670 --> 00:19:28,920 Yep. 334 00:19:28,920 --> 00:19:38,420 Then, am I right that the y derivative of that will be 2y. 335 00:19:38,420 --> 00:19:41,990 That's not looking too good. 336 00:19:41,990 --> 00:19:45,590 What do I want here? 337 00:19:45,590 --> 00:19:49,520 Here's my v, which is the same as w. 338 00:19:49,520 --> 00:19:54,630 And what I'm looking for is to get these guys correct. 339 00:19:54,630 --> 00:19:57,330 So -- and I should be able to do it. 340 00:19:57,330 --> 00:19:59,230 And what would s be? 341 00:19:59,230 --> 00:20:02,410 I haven't got s quite right. 342 00:20:02,410 --> 00:20:04,760 I think if I multiply by negative 1/2, 343 00:20:04,760 --> 00:20:06,490 that might have done it. 344 00:20:06,490 --> 00:20:13,860 Yeah, because now the y derivative is now minus y. 345 00:20:13,860 --> 00:20:15,050 Great. 346 00:20:15,050 --> 00:20:20,410 And the x derivative of s is minus x, 347 00:20:20,410 --> 00:20:22,800 and then I should take a minus that, so I 348 00:20:22,800 --> 00:20:25,890 should want a plus x, which is what I've got. 349 00:20:25,890 --> 00:20:27,950 So those are the streamlines. 350 00:20:27,950 --> 00:20:30,380 Circles. 351 00:20:30,380 --> 00:20:35,530 So I have circle, the flow is going around in a circle. 352 00:20:35,530 --> 00:20:40,550 I don't have to -- I don't need any source to keep it going. 353 00:20:40,550 --> 00:20:48,400 But it's not a gradient. 354 00:20:48,400 --> 00:20:57,760 So this is like a sample test, to take a simple flow field, 355 00:20:57,760 --> 00:21:00,340 apply the two tests, and I guess we 356 00:21:00,340 --> 00:21:05,310 should complete with an example that passes both tests. 357 00:21:05,310 --> 00:21:07,500 Right? 358 00:21:07,500 --> 00:21:09,740 Let me open to any other question, 359 00:21:09,740 --> 00:21:13,460 and then we could cook up an example that passes 360 00:21:13,460 --> 00:21:16,380 both tests before we stop. 361 00:21:16,380 --> 00:21:17,850 I'll stop talking first, though. 362 00:21:17,850 --> 00:21:23,360 Just listen for a question on any topic. 363 00:21:23,360 --> 00:21:26,100 Or is it useful just to take fields like this 364 00:21:26,100 --> 00:21:27,620 and go through those steps? 365 00:21:27,620 --> 00:21:28,610 It probably is. 366 00:21:28,610 --> 00:21:32,020 It's certainly good for me. 367 00:21:32,020 --> 00:21:36,290 OK, what's a field that will satisfy everybody, 368 00:21:36,290 --> 00:21:40,710 that will be a gradient field and also 369 00:21:40,710 --> 00:21:45,030 divergence-free, so that we'll have solutions 370 00:21:45,030 --> 00:21:48,460 to Laplace's equation. 371 00:21:48,460 --> 00:21:49,610 Let's see. 372 00:21:49,610 --> 00:21:59,040 Well we had some solutions to Laplace's equation there. 373 00:21:59,040 --> 00:22:03,880 You know if I make it linear it's real easy. 374 00:22:03,880 --> 00:22:11,450 If I make it quadratic -- huh. 375 00:22:11,450 --> 00:22:15,740 Can I anticipate a little what's coming Friday? 376 00:22:15,740 --> 00:22:20,720 I so recommend to come to Friday's lecture, but -- 377 00:22:20,720 --> 00:22:21,750 so what's coming? 378 00:22:21,750 --> 00:22:23,310 What did we do today? 379 00:22:23,310 --> 00:22:27,760 We discovered that we got solutions to Laplace's equation 380 00:22:27,760 --> 00:22:31,440 from all, by real and imaginary parts of all these guys. 381 00:22:31,440 --> 00:22:33,510 Those were terrific. 382 00:22:33,510 --> 00:22:38,180 And then we could take combinations of those. 383 00:22:38,180 --> 00:22:40,610 So here's what's coming Friday. 384 00:22:40,610 --> 00:22:46,800 When I take combinations of these guys I get some function 385 00:22:46,800 --> 00:22:54,030 of this magic complex -- of this magic combination x+iy. 386 00:22:54,030 --> 00:22:56,940 Some function, any function. 387 00:22:56,940 --> 00:22:59,850 Any reasonable function, and we'll say what reasonable 388 00:22:59,850 --> 00:23:04,820 means, of x+iy, its real part and its imaginary part are 389 00:23:04,820 --> 00:23:06,330 going to be great. 390 00:23:06,330 --> 00:23:10,460 This is like the center of a big, big part of mathematics. 391 00:23:10,460 --> 00:23:11,660 Functions of x+iy. 392 00:23:11,660 --> 00:23:14,560 393 00:23:14,560 --> 00:23:20,030 And by nice I mean that these series converge. 394 00:23:20,030 --> 00:23:22,240 So that we have really a nice function. 395 00:23:22,240 --> 00:23:25,780 Let me take the first function that comes to mind. e^(x+iy). 396 00:23:25,780 --> 00:23:28,510 397 00:23:28,510 --> 00:23:31,850 So let me take this to be e^(x+iy). 398 00:23:31,850 --> 00:23:34,690 399 00:23:34,690 --> 00:23:37,571 OK. 400 00:23:37,571 --> 00:23:38,070 Right. 401 00:23:38,070 --> 00:23:40,590 So you remember, I'm aiming to get solutions 402 00:23:40,590 --> 00:23:43,930 to Laplace's equation, because that will give me automatically 403 00:23:43,930 --> 00:23:46,220 the two pieces both working. 404 00:23:46,220 --> 00:23:50,870 So I claim that the real part of that, and the imaginary part, 405 00:23:50,870 --> 00:23:54,860 those are my twins, u and s, both solve -- 406 00:23:54,860 --> 00:23:59,200 so u is going to be the real part of this function. 407 00:23:59,200 --> 00:24:02,740 And s is going to be the imaginary part of it. 408 00:24:02,740 --> 00:24:06,360 And I claim that those will both solve Laplace's equation. 409 00:24:06,360 --> 00:24:09,150 We can plug it in and see that it does. 410 00:24:09,150 --> 00:24:13,320 And that they will have, they're twinned 411 00:24:13,320 --> 00:24:15,820 by the Cauchy-Riemann equations. 412 00:24:15,820 --> 00:24:18,950 So how am I going to simplify that, 413 00:24:18,950 --> 00:24:22,710 so that I can identify what's the real part of that thing 414 00:24:22,710 --> 00:24:25,170 and what's the imaginary part? 415 00:24:25,170 --> 00:24:30,950 This is actually, that's a good question. 416 00:24:30,950 --> 00:24:36,960 I don't know how much you've run into i, in the past. 417 00:24:36,960 --> 00:24:40,350 Are you happy with something like that? 418 00:24:40,350 --> 00:24:42,360 How could you find the real part of it, 419 00:24:42,360 --> 00:24:44,540 how could you simplify it? 420 00:24:44,540 --> 00:24:51,820 How else could I write e to the something? 421 00:24:51,820 --> 00:24:52,830 Exactly. 422 00:24:52,830 --> 00:24:54,880 Think of it as the product of two, 423 00:24:54,880 --> 00:24:57,790 so the key fact about exponentials is that 424 00:24:57,790 --> 00:25:00,650 that's the same as e^x times e^(iy). 425 00:25:00,650 --> 00:25:03,680 426 00:25:03,680 --> 00:25:07,260 The exponents add, so that's the first thing always 427 00:25:07,260 --> 00:25:09,280 to think about as a possibility. 428 00:25:09,280 --> 00:25:10,950 Now, what am I going to do? 429 00:25:10,950 --> 00:25:13,770 I still want to get a real part. 430 00:25:13,770 --> 00:25:18,330 This is clearly all real, right? e^x is real. 431 00:25:18,330 --> 00:25:22,960 So it's this part that's going to give me the two pieces. 432 00:25:22,960 --> 00:25:25,310 So this is going to be e^x times -- now, 433 00:25:25,310 --> 00:25:28,850 what do I put for e^(iy)? 434 00:25:28,850 --> 00:25:32,820 cos(y), good. 435 00:25:32,820 --> 00:25:35,820 Plus i*sin(y), good. 436 00:25:35,820 --> 00:25:38,090 And now I can read off, no problem. 437 00:25:38,090 --> 00:25:43,990 What is this real part that I was looking for? e^x*cos(y). 438 00:25:43,990 --> 00:25:48,380 439 00:25:48,380 --> 00:25:52,787 And the imaginary part is just what's multiplying the i, 440 00:25:52,787 --> 00:25:53,620 it's the e^x*sin(y). 441 00:25:53,620 --> 00:25:57,840 442 00:25:57,840 --> 00:26:02,080 OK, so what's my claim? 443 00:26:02,080 --> 00:26:05,990 I claim that that function solves Laplace's equation. 444 00:26:05,990 --> 00:26:08,330 And this one too. 445 00:26:08,330 --> 00:26:10,310 And that they're twinned. 446 00:26:10,310 --> 00:26:13,890 And that they give streamlines and equipotentials 447 00:26:13,890 --> 00:26:22,670 that meet at right angles, it's another pair. 448 00:26:22,670 --> 00:26:25,290 Plug that into Laplace's equation. 449 00:26:25,290 --> 00:26:33,835 So let me do u_xx+u_yy, just to satisfy that it is going 450 00:26:33,835 --> 00:26:35,710 to come out zero. 451 00:26:35,710 --> 00:26:39,380 So what's the xx derivative, the second x derivative 452 00:26:39,380 --> 00:26:41,880 of that function? 453 00:26:41,880 --> 00:26:44,620 Take its derivative with respect to x, and then do it again, 454 00:26:44,620 --> 00:26:46,480 and what do you have? 455 00:26:46,480 --> 00:26:47,010 Same. 456 00:26:47,010 --> 00:26:55,330 Didn't change. e^x is just -- and now what about the second y 457 00:26:55,330 --> 00:26:57,060 derivative? 458 00:26:57,060 --> 00:26:59,350 So now e^x is just a constant. 459 00:26:59,350 --> 00:27:01,330 What's the second derivative of cos(y)? 460 00:27:01,330 --> 00:27:03,880 461 00:27:03,880 --> 00:27:05,080 Negative cos(y). 462 00:27:05,080 --> 00:27:05,580 Right. 463 00:27:05,580 --> 00:27:07,870 Because the first derivative is negative sine, 464 00:27:07,870 --> 00:27:10,160 the second derivative is negative cosine. 465 00:27:10,160 --> 00:27:14,300 So the second derivative is e^x, it didn't change, 466 00:27:14,300 --> 00:27:17,440 times cos(y) with a minus sine. 467 00:27:17,440 --> 00:27:23,390 And you see what -- did I write sine? 468 00:27:23,390 --> 00:27:25,710 I meant to write cosine. 469 00:27:25,710 --> 00:27:27,430 Cancel that from the tape. 470 00:27:27,430 --> 00:27:29,171 OK, right. 471 00:27:29,171 --> 00:27:29,670 Yeah. 472 00:27:29,670 --> 00:27:33,065 So the second x derivative was just e^x, e^x, 473 00:27:33,065 --> 00:27:34,900 cos(y) didn't move. 474 00:27:34,900 --> 00:27:35,770 Sorry. 475 00:27:35,770 --> 00:27:37,100 That was frightening. 476 00:27:37,100 --> 00:27:41,850 OK, and then now here's the second y derivative. 477 00:27:41,850 --> 00:27:44,630 In other words, it gives zero. 478 00:27:44,630 --> 00:27:48,760 Gives zero, and this one would too. 479 00:27:48,760 --> 00:28:01,380 Now, I don't really have an idea of what the picture is like. 480 00:28:01,380 --> 00:28:03,890 But it's important. 481 00:28:03,890 --> 00:28:08,780 We've got a flow field here, and it's from e^z, e^(x+iy). 482 00:28:08,780 --> 00:28:12,010 Exponential has gotta be an important function. 483 00:28:12,010 --> 00:28:15,040 So it's got to be somehow interesting. 484 00:28:15,040 --> 00:28:20,140 What do you think -- so what would the -- 485 00:28:20,140 --> 00:28:23,610 what would the equipotential lines looks like? 486 00:28:23,610 --> 00:28:25,760 Oh, boy. 487 00:28:25,760 --> 00:28:29,210 e^x*cos(y) equal a constant. 488 00:28:29,210 --> 00:28:31,740 My gosh. 489 00:28:31,740 --> 00:28:33,350 e^x*cos(y). 490 00:28:33,350 --> 00:28:37,240 So let's see. 491 00:28:37,240 --> 00:28:42,610 I don't know how to draw this picture, but one thing I know 492 00:28:42,610 --> 00:28:48,840 is that if I changed y by 2pi, I would get another copy 493 00:28:48,840 --> 00:28:50,650 of this curve, right? 494 00:28:50,650 --> 00:28:54,060 If I changed y by -- every time you see cosine or sine, 495 00:28:54,060 --> 00:28:55,990 you think hey, that's periodic. 496 00:28:55,990 --> 00:28:57,730 If I change it by 2pi. 497 00:28:57,730 --> 00:29:10,510 So I'm thinking that y between zero and 2 pi, so here's y=0. 498 00:29:10,510 --> 00:29:18,390 And y=2pi, I'm thinking that my flow probably somehow stays 499 00:29:18,390 --> 00:29:20,850 in a strip. 500 00:29:20,850 --> 00:29:21,970 Like that. 501 00:29:21,970 --> 00:29:25,780 And then the whole thing just repeats, and repeats, 502 00:29:25,780 --> 00:29:26,430 and repeats. 503 00:29:26,430 --> 00:29:31,260 So I'm thinking, really this is flow in an infinite strip. 504 00:29:31,260 --> 00:29:33,510 Infinite pipe or something like that. 505 00:29:33,510 --> 00:29:38,160 You can imagine that there could be applications. 506 00:29:38,160 --> 00:29:40,040 But I still haven't drawn the curve. 507 00:29:40,040 --> 00:29:46,080 I just think, let's see, what would it look like when y is 508 00:29:46,080 --> 00:29:50,300 a little -- suppose I'm trying to draw the picture 509 00:29:50,300 --> 00:29:55,380 of e^x*cos(y)=1, whatever. 510 00:29:55,380 --> 00:30:00,970 OK, I'll just attempt to draw that curve. 511 00:30:00,970 --> 00:30:09,180 Just, so if y was a little bit off of zero, 512 00:30:09,180 --> 00:30:14,460 the cosine would be, yeah, how's it going to go? 513 00:30:14,460 --> 00:30:24,980 If y is just a little off zero, tell me 514 00:30:24,980 --> 00:30:30,080 any points on this curve? 515 00:30:30,080 --> 00:30:36,560 I can see that e^x is going to be a big number. 516 00:30:36,560 --> 00:30:39,400 Is (0,0) on the curve? 517 00:30:39,400 --> 00:30:40,140 Good. 518 00:30:40,140 --> 00:30:44,310 Got one point. 519 00:30:44,310 --> 00:30:47,320 Alright. 520 00:30:47,320 --> 00:30:52,120 Now, suppose y is a little bit more than zero. 521 00:30:52,120 --> 00:30:55,190 So suppose y goes up a little bit. 522 00:30:55,190 --> 00:30:59,920 Then what? (1,0) or something? 523 00:30:59,920 --> 00:31:00,420 Yeah. 524 00:31:00,420 --> 00:31:04,530 I suppose (1,0)? 525 00:31:04,530 --> 00:31:08,410 No, no. 526 00:31:08,410 --> 00:31:16,790 So if y goes up a little, then x would go out a little bit. 527 00:31:16,790 --> 00:31:18,080 So what's happening? 528 00:31:18,080 --> 00:31:26,800 So cos(y), so the cos(y) is going to drop from one to zero, 529 00:31:26,800 --> 00:31:28,680 right? 530 00:31:28,680 --> 00:31:30,020 To start with. 531 00:31:30,020 --> 00:31:34,360 Then, if this cos(y) is dropping from one to zero then this e^x 532 00:31:34,360 --> 00:31:39,650 has got to climb up, to to keep the product one. 533 00:31:39,650 --> 00:31:40,530 So I'll move out. 534 00:31:40,530 --> 00:31:42,390 So somehow it'll move out. 535 00:31:42,390 --> 00:31:59,230 I think maybe when y reaches pi/2, 536 00:31:59,230 --> 00:32:06,100 then the cosine has got down to zero. 537 00:32:06,100 --> 00:32:09,270 We could work on this for a while. 538 00:32:09,270 --> 00:32:11,440 Or we could let MATLAB draw it. 539 00:32:11,440 --> 00:32:17,590 But I think that we would see these -- and I could do better. 540 00:32:17,590 --> 00:32:22,900 I'm feeling pretty humiliated to not have a better picture here. 541 00:32:22,900 --> 00:32:25,190 Suppose y is a little less than zero, 542 00:32:25,190 --> 00:32:26,990 do we get anything interesting there? 543 00:32:26,990 --> 00:32:30,610 Oh well, the cosine is an even function. 544 00:32:30,610 --> 00:32:33,740 So I think the thing might, is it just 545 00:32:33,740 --> 00:32:38,600 going to turn around like that? 546 00:32:38,600 --> 00:32:43,450 So that y and minus y -- for a certain x, 547 00:32:43,450 --> 00:32:47,299 the y value and the minus y will both be on the curve 548 00:32:47,299 --> 00:32:49,590 because the cosine doesn't know whether it's the cosine 549 00:32:49,590 --> 00:32:52,170 of of a plus or a minus. 550 00:32:52,170 --> 00:32:58,250 Yeah, I think we would get curves of that sort. 551 00:32:58,250 --> 00:33:04,210 And then the other curves, s equal constant, 552 00:33:04,210 --> 00:33:08,530 the streamlines will somehow go vertically. 553 00:33:08,530 --> 00:33:14,090 Maybe I'll just not use the whole time 554 00:33:14,090 --> 00:33:18,300 to work on that particular curve. 555 00:33:18,300 --> 00:33:20,220 We'd have to prepare it. 556 00:33:20,220 --> 00:33:25,690 The point is, you see how incredibly easily we produce 557 00:33:25,690 --> 00:33:28,450 solutions to Laplace's equation that you 558 00:33:28,450 --> 00:33:31,190 wouldn't have thought of, and I wouldn't have thought of. 559 00:33:31,190 --> 00:33:34,340 So that would be one way to produce solutions. 560 00:33:34,340 --> 00:33:39,410 I might even repeat this one in class Friday, or I might not. 561 00:33:39,410 --> 00:33:43,090 Let me suggest another couple of possibilities 562 00:33:43,090 --> 00:33:45,690 that I will do in class. 563 00:33:45,690 --> 00:33:49,930 Can I just give you a couple of other functions f. 564 00:33:49,930 --> 00:33:54,220 In fact, I'll just erase that one and put in some other ones. 565 00:33:54,220 --> 00:33:58,980 Suppose I took the function 1/(x+iy). 566 00:33:58,980 --> 00:34:07,370 567 00:34:07,370 --> 00:34:11,080 So that's a function of this magic combination, x+iy. 568 00:34:11,080 --> 00:34:15,460 What's its real part and what's its imaginary part? 569 00:34:15,460 --> 00:34:19,470 Do you know how to split that guy into real and imaginary? 570 00:34:19,470 --> 00:34:21,380 There's a little trick, if you remember 571 00:34:21,380 --> 00:34:24,660 from learning complex numbers. 572 00:34:24,660 --> 00:34:26,500 Do you remember the trick? 573 00:34:26,500 --> 00:34:32,130 The problem is that this thing is down in the denominator, 574 00:34:32,130 --> 00:34:33,070 right? 575 00:34:33,070 --> 00:34:34,610 We don't want it there. 576 00:34:34,610 --> 00:34:38,570 Because we can't split the real and imaginary parts down there. 577 00:34:38,570 --> 00:34:41,560 So I would like to rewrite it in a way that 578 00:34:41,560 --> 00:34:45,690 gets something real down in the denominator, 579 00:34:45,690 --> 00:34:48,340 moves all the i stuff up in the numerator 580 00:34:48,340 --> 00:34:50,290 where I can separate it. 581 00:34:50,290 --> 00:34:52,830 How do I do it? 582 00:34:52,830 --> 00:34:59,540 Multiply both sides by, both top and bottom, by x-iy. 583 00:34:59,540 --> 00:35:03,310 Good. 584 00:35:03,310 --> 00:35:05,320 So what does that put down here now? 585 00:35:05,320 --> 00:35:07,140 That's a number times its conjugate 586 00:35:07,140 --> 00:35:11,550 and that's going to produce x squared. 587 00:35:11,550 --> 00:35:13,620 Minus or plus? 588 00:35:13,620 --> 00:35:16,450 Plus y squared, right. 589 00:35:16,450 --> 00:35:19,860 The number times its conjugate is the length squared. 590 00:35:19,860 --> 00:35:27,090 And now we just have x-iy, and now it's obvious what the u is. 591 00:35:27,090 --> 00:35:30,490 This is real now, so the u is just 592 00:35:30,490 --> 00:35:35,550 x over x squared plus y squared, and the s 593 00:35:35,550 --> 00:35:41,930 is the minus y over x squared plus y squared. 594 00:35:41,930 --> 00:35:44,760 That's a very interesting flow. 595 00:35:44,760 --> 00:35:47,070 That's an interesting flow, and we 596 00:35:47,070 --> 00:35:48,800 could do its picture and so on. 597 00:35:48,800 --> 00:35:50,980 And in fact it would be a nicer picture 598 00:35:50,980 --> 00:35:57,400 than the one we stopped on. 599 00:35:57,400 --> 00:36:03,170 What should I notice about this flow? 600 00:36:03,170 --> 00:36:08,760 Of course, the flow is automatically irrotational; 601 00:36:08,760 --> 00:36:13,860 the curl is zero because there is a potential. 602 00:36:13,860 --> 00:36:17,560 A gradient of a potential, the gradient of a potential 603 00:36:17,560 --> 00:36:22,710 is going to be free of rotation. 604 00:36:22,710 --> 00:36:30,550 And there will be streamlines, all those good things. 605 00:36:30,550 --> 00:36:34,090 There's one bad point about the flow, though. 606 00:36:34,090 --> 00:36:36,180 Which is where? 607 00:36:36,180 --> 00:36:37,450 At (0,0). 608 00:36:37,450 --> 00:36:42,690 The whole thing falls apart, at the origin this falls apart. 609 00:36:42,690 --> 00:36:46,240 So this is a great flow except at the origin, 610 00:36:46,240 --> 00:36:49,420 it's very problematic. 611 00:36:49,420 --> 00:36:51,080 It's singular at the origin. 612 00:36:51,080 --> 00:36:56,370 So if we drew the pictures we would see something strange. 613 00:36:56,370 --> 00:36:59,530 This is going to zero at the origin. 614 00:36:59,530 --> 00:37:03,800 So, yeah, we have trouble at the origin but an important flow 615 00:37:03,800 --> 00:37:05,450 otherwise, yep. 616 00:37:05,450 --> 00:37:07,170 And I'll just mention the third but I 617 00:37:07,170 --> 00:37:08,590 won't do anything with it. 618 00:37:08,590 --> 00:37:12,720 Because it's such a neat one that I 619 00:37:12,720 --> 00:37:15,540 have to save it for Friday. 620 00:37:15,540 --> 00:37:19,500 The other natural function to think of is the logarithm. 621 00:37:19,500 --> 00:37:22,890 The logarithm of x+iy. 622 00:37:22,890 --> 00:37:28,870 Split that into u and s. 623 00:37:28,870 --> 00:37:31,320 What kind of a thing do we have here? 624 00:37:31,320 --> 00:37:32,760 What kind of singularity? 625 00:37:32,760 --> 00:37:38,080 Yeah, let me just do two moments on this example, 626 00:37:38,080 --> 00:37:41,560 and then leave it for Friday because the whole class has 627 00:37:41,560 --> 00:37:44,090 to see it. 628 00:37:44,090 --> 00:37:47,530 Is there a singularity for this guy? 629 00:37:47,530 --> 00:37:56,300 Is there a point (x,y) where the logarithm is not great? 630 00:37:56,300 --> 00:37:59,460 At the origin, again. 631 00:37:59,460 --> 00:38:01,660 We'll again have a singularity at the origin. 632 00:38:01,660 --> 00:38:05,260 Something strange is happening at the origin. 633 00:38:05,260 --> 00:38:08,730 And what we'll find is there's a delta function there. 634 00:38:08,730 --> 00:38:14,220 We're feeding in, we have a source right at the origin 635 00:38:14,220 --> 00:38:20,410 and then it's flowing out on, I think on radial lines. 636 00:38:20,410 --> 00:38:23,600 I think the streamlines go out from the origin 637 00:38:23,600 --> 00:38:26,180 and the equipotentials go around the origin. 638 00:38:26,180 --> 00:38:28,220 Yeah, it's a great example. 639 00:38:28,220 --> 00:38:31,200 So that's another one to come. 640 00:38:31,200 --> 00:38:34,290 OK, so examples like these are -- 641 00:38:34,290 --> 00:38:41,820 I mean generations of thinking went into solutions 642 00:38:41,820 --> 00:38:44,380 of Laplace's equation. 643 00:38:44,380 --> 00:38:50,550 And 2-D particularly where we have this special combination. 644 00:38:50,550 --> 00:38:55,210 I wish we had such a combination in 3-D but we simply don't. 645 00:38:55,210 --> 00:38:58,780 We can discuss Laplace's equation in 3-D of course, 646 00:38:58,780 --> 00:38:59,590 very important. 647 00:38:59,590 --> 00:39:03,980 But I mean, wave equation, this fact that we're talking to each 648 00:39:03,980 --> 00:39:16,520 other, is got the Laplacian in 3-D, but there's no x+iy magic. 649 00:39:16,520 --> 00:39:21,170 OK, that's some u's and s's and v's and w's. 650 00:39:21,170 --> 00:39:25,880 What else is on your mind? 651 00:39:25,880 --> 00:39:30,420 Questions? 652 00:39:30,420 --> 00:39:33,200 I could ask this question, oh, here's something 653 00:39:33,200 --> 00:39:37,910 I did not do in class. 654 00:39:37,910 --> 00:39:40,070 I think I wrote down the divergence theorem. 655 00:39:40,070 --> 00:39:42,380 So can we start by doing that? 656 00:39:42,380 --> 00:39:47,350 Let me write down the divergence theorem, with your help. 657 00:39:47,350 --> 00:39:55,270 And then use it. 658 00:39:55,270 --> 00:39:58,280 So what does the divergence theorem -- we're in 2-D. 659 00:39:58,280 --> 00:40:03,230 So this is 2-D, just the similar theorem. 660 00:40:03,230 --> 00:40:08,950 So what does the theorem say, that if I take the divergence 661 00:40:08,950 --> 00:40:19,710 of some w, some vector field, then if I integrate that over 662 00:40:19,710 --> 00:40:24,370 some region -- so I have some region here and at every point 663 00:40:24,370 --> 00:40:26,950 there's a flow w. 664 00:40:26,950 --> 00:40:33,700 And I look at the divergence of w and I integrate that, dx dy, 665 00:40:33,700 --> 00:40:37,500 so that's a double integral over a region, 666 00:40:37,500 --> 00:40:44,950 I will get, what's the right-hand side? 667 00:40:44,950 --> 00:40:48,040 What does the divergence measure? 668 00:40:48,040 --> 00:40:53,380 So I'm really asking like just memory, what is the divergence, 669 00:40:53,380 --> 00:40:54,740 it's an identity. 670 00:40:54,740 --> 00:41:00,990 It's integration by parts in some way, as we'll see. 671 00:41:00,990 --> 00:41:06,470 But what do you remember for the divergence theorem? 672 00:41:06,470 --> 00:41:09,070 You get what? 673 00:41:09,070 --> 00:41:14,020 It measures how much flux out, right? 674 00:41:14,020 --> 00:41:16,750 So when we measure the flux out by integrating 675 00:41:16,750 --> 00:41:21,700 around the boundary, how much is getting through the boundary? 676 00:41:21,700 --> 00:41:24,870 And what's the flow through the boundary? 677 00:41:24,870 --> 00:41:28,430 I take w, but that's a vector. 678 00:41:28,430 --> 00:41:32,340 And I'm looking for what component of w? 679 00:41:32,340 --> 00:41:36,230 The normal component, the component of w, w dot n, 680 00:41:36,230 --> 00:41:42,040 the component of w that's headed out. n is defined to be, 681 00:41:42,040 --> 00:41:46,100 whatever the boundary is -- here I've made it look like a circle 682 00:41:46,100 --> 00:41:47,650 but I shouldn't have. 683 00:41:47,650 --> 00:41:51,730 Let me make it a little wobblier or something. 684 00:41:51,730 --> 00:41:57,990 So the normal component at any, there, look at that point. 685 00:41:57,990 --> 00:42:01,630 The normal direction through the boundary, 686 00:42:01,630 --> 00:42:05,390 down in that crazy point, is this way. 687 00:42:05,390 --> 00:42:08,090 So the normal is going this way here. 688 00:42:08,090 --> 00:42:10,100 Here, it's going over this way. 689 00:42:10,100 --> 00:42:13,680 It's perpendicular to the boundary, OK? 690 00:42:13,680 --> 00:42:17,160 And then we integrate around the boundary. 691 00:42:17,160 --> 00:42:20,880 Alright. 692 00:42:20,880 --> 00:42:27,440 So that's the identity of that, that's the divergence theorem. 693 00:42:27,440 --> 00:42:32,280 Now, let's see. 694 00:42:32,280 --> 00:42:33,510 Could you, yeah. 695 00:42:33,510 --> 00:42:38,260 So we have a minute. 696 00:42:38,260 --> 00:42:41,990 You want to take a particular w and see 697 00:42:41,990 --> 00:42:45,770 if this would be correct? 698 00:42:45,770 --> 00:42:50,710 How about w=w=[0, x], our first example? 699 00:42:50,710 --> 00:42:52,830 Suppose I tried w=[0, x]. 700 00:42:52,830 --> 00:42:55,640 I just want to see if the divergence -- 701 00:42:55,640 --> 00:42:58,980 what the flux is through the boundary. 702 00:42:58,980 --> 00:43:03,130 What what region shall I take for the -- 703 00:43:03,130 --> 00:43:05,470 so the divergence theorem has two inputs. 704 00:43:05,470 --> 00:43:07,320 It has a flow field. 705 00:43:07,320 --> 00:43:11,580 And let me take w to be [0, x], just so it's a shear. 706 00:43:11,580 --> 00:43:14,260 And a region. 707 00:43:14,260 --> 00:43:18,650 And of course the integral might not be that much fun to do, 708 00:43:18,650 --> 00:43:21,010 unless we make the region nice. 709 00:43:21,010 --> 00:43:29,900 What do you take as a nice region for -- actually, 710 00:43:29,900 --> 00:43:32,320 it doesn't matter what the region is. 711 00:43:32,320 --> 00:43:33,510 Take any old region. 712 00:43:33,510 --> 00:43:36,580 For the moment. 713 00:43:36,580 --> 00:43:38,570 What's the answer? 714 00:43:38,570 --> 00:43:41,040 For this particular flow, w=w=[0, x]? 715 00:43:41,040 --> 00:43:45,870 716 00:43:45,870 --> 00:43:48,550 Zero. 717 00:43:48,550 --> 00:43:49,570 That's the cool part. 718 00:43:49,570 --> 00:43:54,500 If the answer's zero then work is suspended. 719 00:43:54,500 --> 00:43:56,840 And why is it zero? 720 00:43:56,840 --> 00:44:00,620 Because the divergence of this particular w, 721 00:44:00,620 --> 00:44:03,550 the x derivative of that plus the y derivative of that, 722 00:44:03,550 --> 00:44:04,050 is zero. 723 00:44:04,050 --> 00:44:07,800 This has divergence everywhere zero. 724 00:44:07,800 --> 00:44:10,740 So integrating is no problem at all, 725 00:44:10,740 --> 00:44:15,000 so that would be zero, for this flow field. 726 00:44:15,000 --> 00:44:17,290 For this divergence-free field. 727 00:44:17,290 --> 00:44:24,610 Zero for that because div w is zero. 728 00:44:24,610 --> 00:44:26,290 But is that correct? 729 00:44:26,290 --> 00:44:29,880 What does that tell me, these flows are -- 730 00:44:29,880 --> 00:44:33,870 we saw what the flow is like. 731 00:44:33,870 --> 00:44:35,990 Say there's the origin. 732 00:44:35,990 --> 00:44:40,230 It doesn't have to be a circle, it looks like a circle. 733 00:44:40,230 --> 00:44:42,700 Do you see why the flux is zero? 734 00:44:42,700 --> 00:44:46,540 There is flow through the boundary, right? 735 00:44:46,540 --> 00:44:52,130 Flow is going buzz, buzz, buzz up this line and out. 736 00:44:52,130 --> 00:44:54,920 And it's coming in here. 737 00:44:54,920 --> 00:44:57,740 So there that's what we discovered. 738 00:44:57,740 --> 00:45:04,080 This [0, x] is vertical flow. 739 00:45:04,080 --> 00:45:07,750 It hasn't gotten any horizontal component. 740 00:45:07,750 --> 00:45:10,320 It's got a vertical component, it's going out. 741 00:45:10,320 --> 00:45:13,940 And would we want to do this right-hand side? 742 00:45:13,940 --> 00:45:16,670 I don't think so, right. 743 00:45:16,670 --> 00:45:18,370 This right-hand side is asking me 744 00:45:18,370 --> 00:45:21,960 what, I have to find the normal direction on this, whatever 745 00:45:21,960 --> 00:45:23,650 curve that is. 746 00:45:23,650 --> 00:45:28,020 I have to take its dot product with the flow [0, x], 747 00:45:28,020 --> 00:45:33,180 so this is some quantity. 748 00:45:33,180 --> 00:45:36,400 And then I have to do this ds which I haven't even mentioned, 749 00:45:36,400 --> 00:45:38,350 ds is arc length around. 750 00:45:38,350 --> 00:45:40,680 I'm integrating around these pieces. 751 00:45:40,680 --> 00:45:48,690 But yet somehow we have some idea from that picture 752 00:45:48,690 --> 00:45:52,670 that the total flux is zero. 753 00:45:52,670 --> 00:45:57,210 How would you say it in words, if I say here's the flow field. 754 00:45:57,210 --> 00:46:01,690 There's a region, funny shape. 755 00:46:01,690 --> 00:46:03,880 The flux is zero through that boundary. 756 00:46:03,880 --> 00:46:07,700 And if I asked you why, what would you say? 757 00:46:07,700 --> 00:46:11,690 I mean, a math answer would be use the divergence theorem. 758 00:46:11,690 --> 00:46:19,300 But why from this picture does it look like we have zero flux? 759 00:46:19,300 --> 00:46:21,200 What comes in goes out, yeah. 760 00:46:21,200 --> 00:46:25,840 What's coming in the bottom here is going out the top. 761 00:46:25,840 --> 00:46:29,620 That's basically it. 762 00:46:29,620 --> 00:46:34,620 So we would get zero for that one. 763 00:46:34,620 --> 00:46:38,260 So I think the homework, the suggested homework 764 00:46:38,260 --> 00:46:42,430 maybe includes an example where the divergence isn't zero. 765 00:46:42,430 --> 00:46:44,910 And then you actually have to do these integrals. 766 00:46:44,910 --> 00:46:50,450 Just as practice for what do those integrals mean. 767 00:46:50,450 --> 00:46:55,690 Maybe I won't go through one now, but that's good practice. 768 00:46:55,690 --> 00:47:03,860 Take some simple w, but one with a non-zero divergence 769 00:47:03,860 --> 00:47:06,770 and then see if you can do either or both 770 00:47:06,770 --> 00:47:11,500 of the integrals that are supposed to come out equal. 771 00:47:11,500 --> 00:47:15,970 That's a good one Now, there's one thing I could -- 772 00:47:15,970 --> 00:47:18,990 any questions, or discussion? 773 00:47:18,990 --> 00:47:25,090 You guys are seeing these examples come up; 774 00:47:25,090 --> 00:47:28,080 it's the only way I would know to learn this subject is take 775 00:47:28,080 --> 00:47:30,210 simple v's and w's. 776 00:47:30,210 --> 00:47:36,140 And see what you can do with them. 777 00:47:36,140 --> 00:47:37,800 We've got the general principles, 778 00:47:37,800 --> 00:47:42,300 but then apply them to specific flows. 779 00:47:42,300 --> 00:47:44,920 AUDIENCE: [INAUDIBLE] PROFESSOR STRANG: Yes, thanks 780 00:47:44,920 --> 00:47:51,729 AUDIENCE: [INAUDIBLE] 781 00:47:51,729 --> 00:47:53,020 PROFESSOR STRANG: This theorem? 782 00:47:53,020 --> 00:47:58,356 AUDIENCE: [INAUDIBLE] 783 00:47:58,356 --> 00:47:59,730 PROFESSOR STRANG: Yeah if it was, 784 00:47:59,730 --> 00:48:02,350 well let's draw a funny shape. 785 00:48:02,350 --> 00:48:03,680 See what we think. 786 00:48:03,680 --> 00:48:11,470 I mean, with this flow, right? 787 00:48:11,470 --> 00:48:16,510 Let me just say, if the flow has some difficult divergence 788 00:48:16,510 --> 00:48:19,550 and the region is some mess, nobody's 789 00:48:19,550 --> 00:48:20,810 going to be able to do it. 790 00:48:20,810 --> 00:48:21,770 I mean, yeah. 791 00:48:21,770 --> 00:48:24,460 So don't think that these can all be done. 792 00:48:24,460 --> 00:48:29,150 The equality, it's like integrals in calculus. 793 00:48:29,150 --> 00:48:31,020 No problem to think of integrations that 794 00:48:31,020 --> 00:48:33,620 are just beyond human capacity. 795 00:48:33,620 --> 00:48:36,680 But the formulas still hold. 796 00:48:36,680 --> 00:48:43,510 Suppose my region was even like this? 797 00:48:43,510 --> 00:48:48,340 Would that still be, do we still see flow in equals flow out 798 00:48:48,340 --> 00:48:50,320 for this particular flow? 799 00:48:50,320 --> 00:48:53,780 I think, yeah, the flow's going this way. 800 00:48:53,780 --> 00:48:56,430 So it's coming in here, it's going out again. 801 00:48:56,430 --> 00:48:57,790 That contributes. 802 00:48:57,790 --> 00:49:00,660 Back in again here, and out again here. 803 00:49:00,660 --> 00:49:09,290 Yeah, I think our instinct would be correct there, yeah. 804 00:49:09,290 --> 00:49:11,420 All sorts of examples. 805 00:49:11,420 --> 00:49:16,890 I was going to, well, I'll maybe do it in class. 806 00:49:16,890 --> 00:49:20,750 This divergence theorem is the fundamental theorem 807 00:49:20,750 --> 00:49:22,920 of 2-D calculus, you could say. 808 00:49:22,920 --> 00:49:27,930 Or one of them. 809 00:49:27,930 --> 00:49:42,050 And to write these things and see what they lead to, yeah. 810 00:49:42,050 --> 00:49:44,500 I'll tell you what I was going to do. 811 00:49:44,500 --> 00:49:52,220 I was going to apply this to the vector field u times w. 812 00:49:52,220 --> 00:49:58,180 So u is a scalar, w is a vector, and therefore uw is a vector. 813 00:49:58,180 --> 00:50:06,230 It's got two components, uw_1, are you willing to do that one? 814 00:50:06,230 --> 00:50:16,820 So I apply this not to w itself, but to u times w. 815 00:50:16,820 --> 00:50:22,720 Which has two components, uw_1 and uw_2. 816 00:50:22,720 --> 00:50:25,460 OK, so I should take the divergence of uw. 817 00:50:25,460 --> 00:50:31,240 I mean, that's a vector field, uw, this guy. 818 00:50:31,240 --> 00:50:35,420 And I'll get the uw dot n. 819 00:50:35,420 --> 00:50:41,700 And it just turns out that this is the right way to do it. 820 00:50:41,700 --> 00:50:44,750 To see the fact that gradient and divergence 821 00:50:44,750 --> 00:50:47,070 are transposes of each other. 822 00:50:47,070 --> 00:50:48,100 Yeah, yeah. 823 00:50:48,100 --> 00:50:51,060 I maybe I won't do that calculation now, 824 00:50:51,060 --> 00:50:55,790 I'll just say that if you take the divergence theorem and you 825 00:50:55,790 --> 00:51:01,290 apply it to this guy, [uw 1, uw 2], 826 00:51:01,290 --> 00:51:10,190 and write out what it means, you get a very interesting formula. 827 00:51:10,190 --> 00:51:12,030 I'll just leave that there. 828 00:51:12,030 --> 00:51:14,570 So I'm ready for a final question if there is one, 829 00:51:14,570 --> 00:51:21,940 or otherwise keep going Friday with these Laplace equation 830 00:51:21,940 --> 00:51:23,210 solutions. 831 00:51:23,210 --> 00:51:25,780 Play with some vector fields. 832 00:51:25,780 --> 00:51:29,740 That's my best advice. 833 00:51:29,740 --> 00:51:34,090 And I'll see you Friday.