1 00:00:00,000 --> 00:00:00,027 2 00:00:00,027 --> 00:00:02,110 The following content is provided under a Creative 3 00:00:02,110 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,970 Your support will help MIT OpenCourseWare 5 00:00:05,970 --> 00:00:10,050 continue to offer high-quality educational resources for free. 6 00:00:10,050 --> 00:00:12,530 To make a donation, or to view additional materials 7 00:00:12,530 --> 00:00:16,880 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,880 --> 00:00:19,390 at ocw.mit.edu. 9 00:00:19,390 --> 00:00:22,590 PROFESSOR STRANG: OK. 10 00:00:22,590 --> 00:00:24,430 So this is a fun lecture. 11 00:00:24,430 --> 00:00:27,640 This is the lecture where understanding 12 00:00:27,640 --> 00:00:30,460 the gradient, which we did last time; 13 00:00:30,460 --> 00:00:33,250 the divergence, that we almost completed; 14 00:00:33,250 --> 00:00:35,630 those two pieces now come together 15 00:00:35,630 --> 00:00:37,870 into Laplace's equation. 16 00:00:37,870 --> 00:00:42,150 Things work in a fantastic way, so I enjoy this one. 17 00:00:42,150 --> 00:00:47,440 So the result will be that we have the potential function 18 00:00:47,440 --> 00:00:52,910 u that we've spoken about, and take its gradient. 19 00:00:52,910 --> 00:00:56,580 Now is coming a stream function, s, that's 20 00:00:56,580 --> 00:00:58,460 connected with the divergence. 21 00:00:58,460 --> 00:01:01,220 When the divergence is zero, there's 22 00:01:01,220 --> 00:01:03,510 something called a stream function, s. 23 00:01:03,510 --> 00:01:07,540 And these, the connections between those two functions, u 24 00:01:07,540 --> 00:01:11,730 and s, are crucial. 25 00:01:11,730 --> 00:01:15,810 And connecting gradient to divergence 26 00:01:15,810 --> 00:01:18,620 will take us to Laplace's equation. 27 00:01:18,620 --> 00:01:25,460 And then you'll see the special, special role of x+iy in 2-D. 28 00:01:25,460 --> 00:01:27,680 So this is in two dimensions. 29 00:01:27,680 --> 00:01:28,270 OK. 30 00:01:28,270 --> 00:01:32,290 So let me begin with divergence. 31 00:01:32,290 --> 00:01:36,460 This is the divergence of w, of course. 32 00:01:36,460 --> 00:01:40,030 And solve it. 33 00:01:40,030 --> 00:01:42,700 Just as we wanted to solve Kirchhoff's current law, 34 00:01:42,700 --> 00:01:46,840 A transpose w equal zero, so now in the continuous case, 35 00:01:46,840 --> 00:01:53,710 we want to find solutions to -- we want to find divergence-free 36 00:01:53,710 --> 00:01:54,580 fields. 37 00:01:54,580 --> 00:01:57,100 Source-free fields, you could say. 38 00:01:57,100 --> 00:01:58,720 That's about the best word we have, 39 00:01:58,720 --> 00:02:05,020 divergence-free, meaning there is no divergence. 40 00:02:05,020 --> 00:02:07,960 So what have we got here? 41 00:02:07,960 --> 00:02:12,070 We've got one equation in two unknowns. 42 00:02:12,070 --> 00:02:14,170 Last time, with the gradient business, 43 00:02:14,170 --> 00:02:18,180 we had two equations for u, and only that one unknown. 44 00:02:18,180 --> 00:02:20,870 Now we've got one equation, two unknowns, because we're 45 00:02:20,870 --> 00:02:23,400 looking at the transpose. 46 00:02:23,400 --> 00:02:25,040 So there should be a lot of solutions. 47 00:02:25,040 --> 00:02:26,370 Right? 48 00:02:26,370 --> 00:02:29,750 If you give me a w_1, then probably 49 00:02:29,750 --> 00:02:31,830 I'll be able to find a w_2. 50 00:02:31,830 --> 00:02:35,210 But there's a neat way to describe the solutions 51 00:02:35,210 --> 00:02:38,280 to that equation, the divergence-free fields. 52 00:02:38,280 --> 00:02:41,550 It's to introduce something called a source function -- 53 00:02:41,550 --> 00:02:44,980 a stream function, sorry, stream function. 54 00:02:44,980 --> 00:02:53,640 OK, and now the idea will be that if I let -- let me try. 55 00:02:53,640 --> 00:02:57,860 I take any function s(x,y), any function whatever. 56 00:02:57,860 --> 00:03:04,920 So now I try -- let w_1 be ds/dy. 57 00:03:04,920 --> 00:03:08,050 58 00:03:08,050 --> 00:03:14,560 Maybe I should say, maybe I should go this way. 59 00:03:14,560 --> 00:03:16,420 I'm looking for solutions. 60 00:03:16,420 --> 00:03:16,970 OK. 61 00:03:16,970 --> 00:03:23,010 So I take any function s, I take its y derivative to be w_1. 62 00:03:23,010 --> 00:03:28,340 And you can tell me what w_2 has to be? 63 00:03:28,340 --> 00:03:33,280 So if w_1 is ds/dy, this will be what? 64 00:03:33,280 --> 00:03:38,260 This'll be the second -- the x derivative of the y derivative. 65 00:03:38,260 --> 00:03:39,480 Right? 66 00:03:39,480 --> 00:03:43,590 If w_1 is the y derivative, then when I take the x derivative, 67 00:03:43,590 --> 00:03:45,470 I've got this cross derivative. 68 00:03:45,470 --> 00:03:49,040 So what would be the smart choice for w_2? 69 00:03:49,040 --> 00:03:55,510 So I'll say then, then w_2 will be -- 70 00:03:55,510 --> 00:03:58,720 now I've sort of said what w_1 is, 71 00:03:58,720 --> 00:04:01,230 what's the w_2 that goes with it? 72 00:04:01,230 --> 00:04:04,840 Well, it's whatever it takes to cancel this. 73 00:04:04,840 --> 00:04:08,530 I want this to be, in other words I want it to get zero, 74 00:04:08,530 --> 00:04:12,050 so this should be the second derivative s. 75 00:04:12,050 --> 00:04:16,400 And what am I going to put now? dydx. 76 00:04:16,400 --> 00:04:20,510 I'm using the same very crucial fact 77 00:04:20,510 --> 00:04:23,420 that the cross derivative can be in either order. 78 00:04:23,420 --> 00:04:26,770 So what do I see for w_2? 79 00:04:26,770 --> 00:04:28,230 Do you see what w_2 is? 80 00:04:28,230 --> 00:04:30,610 This is supposed to match that. 81 00:04:30,610 --> 00:04:35,610 There's a minus sign. w_2 is what? 82 00:04:35,610 --> 00:04:36,650 Minus ds/dx. 83 00:04:36,650 --> 00:04:39,260 84 00:04:39,260 --> 00:04:39,760 Right? 85 00:04:39,760 --> 00:04:42,320 Minus ds/dx. 86 00:04:42,320 --> 00:04:46,060 That's the minus ds/dx. 87 00:04:46,060 --> 00:04:50,250 If I take any function s, I let w_1 be its y derivative 88 00:04:50,250 --> 00:04:53,890 and w_2 be minus its x derivative, 89 00:04:53,890 --> 00:04:57,030 then the divergence will be zero. 90 00:04:57,030 --> 00:04:59,550 Because I'll have the cross derivative 91 00:04:59,550 --> 00:05:02,400 minus the cross derivative, of course that'll be zero. 92 00:05:02,400 --> 00:05:08,720 So these will be my w_1's -- my w's. w_1, w_2, coming from s. 93 00:05:08,720 --> 00:05:12,300 So I have what I expect, with two unknowns, 94 00:05:12,300 --> 00:05:15,090 only one equation, I've got lots of solutions. 95 00:05:15,090 --> 00:05:18,950 I create any function s, and that will be one. 96 00:05:18,950 --> 00:05:21,570 So that's a stream function. 97 00:05:21,570 --> 00:05:24,640 And it has a physical meaning, so we get to see it. 98 00:05:24,640 --> 00:05:30,000 And it has a fantastic connection to the potential. 99 00:05:30,000 --> 00:05:33,830 So up to now -- so this is now the moment pieces come 100 00:05:33,830 --> 00:05:35,370 together. 101 00:05:35,370 --> 00:05:39,470 Up to now, we had the divergence and the gradient separately. 102 00:05:39,470 --> 00:05:42,680 So up to now, what we had was, we started with the potential 103 00:05:42,680 --> 00:05:49,375 u, and we went to v -- I called it v for this application -- 104 00:05:49,375 --> 00:05:50,000 [du/dx, du/dy]. 105 00:05:50,000 --> 00:05:54,010 106 00:05:54,010 --> 00:05:54,890 OK. 107 00:05:54,890 --> 00:05:58,590 And now, over on this side, I had the divergence of w 108 00:05:58,590 --> 00:06:00,230 equals zero. 109 00:06:00,230 --> 00:06:08,670 And that led me to w being, what we just said, ds/dy, 110 00:06:08,670 --> 00:06:11,710 and minus ds/dx. 111 00:06:11,710 --> 00:06:15,220 OK, two separate pictures. 112 00:06:15,220 --> 00:06:19,480 Now we're going to connect the framework, 113 00:06:19,480 --> 00:06:23,160 going to give the connection between v and w. 114 00:06:23,160 --> 00:06:25,560 And the connection will be the easiest possible; 115 00:06:25,560 --> 00:06:26,820 they'll be equal. 116 00:06:26,820 --> 00:06:29,870 So the c in our framework is the identity. 117 00:06:29,870 --> 00:06:33,290 I want to say, I want to look at our framework 118 00:06:33,290 --> 00:06:36,830 when c is the identity. 119 00:06:36,830 --> 00:06:39,070 So v and w are the same. 120 00:06:39,070 --> 00:06:41,730 In other words, what equation do we get then? 121 00:06:41,730 --> 00:06:44,440 We started with u, we take the gradient 122 00:06:44,440 --> 00:06:48,760 of u, that's v. We go over here, and we still 123 00:06:48,760 --> 00:06:53,720 have the gradient of u, because this was the v 124 00:06:53,720 --> 00:06:55,770 and now it's also the w. 125 00:06:55,770 --> 00:06:58,160 And then we take -- what do we take? 126 00:06:58,160 --> 00:07:03,390 Minus the divergence of the gradient of u, equals, 127 00:07:03,390 --> 00:07:04,510 let's say, f. 128 00:07:04,510 --> 00:07:07,740 Shall we say f, to make something happen? 129 00:07:07,740 --> 00:07:12,570 Or zero -- I'll say zero. 130 00:07:12,570 --> 00:07:14,300 I'll say zero. 131 00:07:14,300 --> 00:07:16,330 Yeah, because zero is the one that 132 00:07:16,330 --> 00:07:18,490 give us Laplace's equation. 133 00:07:18,490 --> 00:07:22,200 The official name would be -- that's Laplace's equation. 134 00:07:22,200 --> 00:07:25,120 Do you recognize it? 135 00:07:25,120 --> 00:07:27,980 We'd better write it out properly. 136 00:07:27,980 --> 00:07:31,970 The divergence of the gradient is the Laplacian. 137 00:07:31,970 --> 00:07:35,600 So this is Laplace. 138 00:07:35,600 --> 00:07:38,560 Famous equation. 139 00:07:38,560 --> 00:07:40,510 Steady-state type of equation. 140 00:07:40,510 --> 00:07:42,630 It goes with steady-state problems. 141 00:07:42,630 --> 00:07:45,950 Let's just figure out what this is. 142 00:07:45,950 --> 00:07:55,420 So w is, since these are the same, w is now [du/dx, du/dy]. 143 00:07:55,420 --> 00:07:56,690 It's the gradient. 144 00:07:56,690 --> 00:07:58,200 Now what happens, do you see what 145 00:07:58,200 --> 00:08:01,290 happens when I take the divergence of the gradient, 146 00:08:01,290 --> 00:08:07,210 the divergence of this w? 147 00:08:07,210 --> 00:08:09,390 I just want to write that equation out. 148 00:08:09,390 --> 00:08:12,460 It's crucial. 149 00:08:12,460 --> 00:08:17,200 I need brilliant colors and lights shining, now, 150 00:08:17,200 --> 00:08:19,460 for what goes in there. 151 00:08:19,460 --> 00:08:22,100 Because I want to take the divergence of the gradient. 152 00:08:22,100 --> 00:08:23,320 So what does that mean? 153 00:08:23,320 --> 00:08:26,460 I take the x derivative of the first component, 154 00:08:26,460 --> 00:08:29,190 plus the y derivative of the second component. 155 00:08:29,190 --> 00:08:32,530 The minus sign is not going to matter, with a zero 156 00:08:32,530 --> 00:08:34,140 on the right hand side. 157 00:08:34,140 --> 00:08:35,850 I could cancel the minus. 158 00:08:35,850 --> 00:08:38,700 So, but let me say it again. 159 00:08:38,700 --> 00:08:43,120 The divergence is, it applies to a vector, I've got a vector. 160 00:08:43,120 --> 00:08:47,940 I take the x component of the first -- of w_1. 161 00:08:47,940 --> 00:08:52,100 So that'll give me d second u/dx squared. 162 00:08:52,100 --> 00:08:56,230 And I take the y derivative -- I should have said derivative -- 163 00:08:56,230 --> 00:08:59,340 y derivative of the second component. 164 00:08:59,340 --> 00:09:05,570 And the y derivative of that is d second u/dy squared. 165 00:09:05,570 --> 00:09:08,030 And I get zero. 166 00:09:08,030 --> 00:09:11,330 So that's Laplace's equation. 167 00:09:11,330 --> 00:09:15,200 You see, the whole idea is Laplace's equation, 168 00:09:15,200 --> 00:09:17,300 in working with Laplace's equation, 169 00:09:17,300 --> 00:09:20,930 we have three elements, here. 170 00:09:20,930 --> 00:09:25,430 The gradient comes in, the divergence comes in, 171 00:09:25,430 --> 00:09:29,840 and equality comes in. 172 00:09:29,840 --> 00:09:33,230 Would you like to see a more general Laplace's equation? 173 00:09:33,230 --> 00:09:35,610 Well, a more general Poisson's equation. 174 00:09:35,610 --> 00:09:43,080 So underneath Laplace, let me write Poisson. 175 00:09:43,080 --> 00:09:46,440 You got the pronunciation, the brilliant French pronunciation 176 00:09:46,440 --> 00:09:49,080 there, of Poisson? 177 00:09:49,080 --> 00:09:50,130 That was my best. 178 00:09:50,130 --> 00:09:52,620 I can't improve on that one. 179 00:09:52,620 --> 00:09:55,730 So it comes in when there's a right-hand side. 180 00:09:55,730 --> 00:09:59,920 So the normal Poisson equation is second derivative with 181 00:09:59,920 --> 00:10:03,680 respect to x, second derivative with respect to y -- well, 182 00:10:03,680 --> 00:10:07,260 actually, it should have a minus there. 183 00:10:07,260 --> 00:10:11,050 Really should be a minus there. 184 00:10:11,050 --> 00:10:13,090 You remember why the minus is there. 185 00:10:13,090 --> 00:10:15,840 The minus is there to make the whole thing positive. 186 00:10:15,840 --> 00:10:16,340 Right? 187 00:10:16,340 --> 00:10:19,190 That sounds crazy, that the minus makes it positive. 188 00:10:19,190 --> 00:10:23,790 But these second derivatives are negative definite, as always, 189 00:10:23,790 --> 00:10:26,370 and the minus makes them positive definite. 190 00:10:26,370 --> 00:10:32,350 So I don't remember whether -- maybe I often include the minus 191 00:10:32,350 --> 00:10:35,770 over here -- equal f(x,y). 192 00:10:35,770 --> 00:10:36,920 So there's a source. 193 00:10:36,920 --> 00:10:38,740 Poisson has a source term. 194 00:10:38,740 --> 00:10:41,950 Laplace doesn't. 195 00:10:41,950 --> 00:10:44,760 And just while I'm talking about this framework, 196 00:10:44,760 --> 00:10:49,800 if there was a c(x,y), so this would be or. 197 00:10:49,800 --> 00:10:51,980 I'll put or. 198 00:10:51,980 --> 00:10:55,450 The more general one would be the x derivative, 199 00:10:55,450 --> 00:11:03,410 because I'm taking the divergence, of a c(x,y)du/dx, 200 00:11:03,410 --> 00:11:08,140 and the y derivative of a c -- so you see the difference. 201 00:11:08,140 --> 00:11:13,130 I'm now allowing some variable conductivity. 202 00:11:13,130 --> 00:11:18,010 Variable whatever, variable material. 203 00:11:18,010 --> 00:11:22,490 du/dy equals f, again. 204 00:11:22,490 --> 00:11:24,470 So that would be the more general one. 205 00:11:24,470 --> 00:11:27,290 I don't think we plan to study that. 206 00:11:27,290 --> 00:11:29,280 Well, that's not the most general, 207 00:11:29,280 --> 00:11:32,110 I could have more and more things there. 208 00:11:32,110 --> 00:11:36,880 But that shows you a variable material. 209 00:11:36,880 --> 00:11:39,180 Yeah, that material is variable. 210 00:11:39,180 --> 00:11:42,130 I would use the word -- what word would I use? 211 00:11:42,130 --> 00:11:44,600 It doesn't depend on the direction. 212 00:11:44,600 --> 00:11:48,760 I'm using the same c(x,y) in the x direction and the y 213 00:11:48,760 --> 00:11:49,880 direction. 214 00:11:49,880 --> 00:11:51,390 Therefore in all directions. 215 00:11:51,390 --> 00:11:53,820 And I didn't come prepared with that word. 216 00:11:53,820 --> 00:11:56,290 What's the word for when it doesn't 217 00:11:56,290 --> 00:11:58,890 depend on the direction? 218 00:11:58,890 --> 00:11:59,740 Isotropic! 219 00:11:59,740 --> 00:12:00,340 Thanks. 220 00:12:00,340 --> 00:12:01,330 Isotropic. 221 00:12:01,330 --> 00:12:03,320 So that's isotropic. 222 00:12:03,320 --> 00:12:07,230 So that's really our framework there. 223 00:12:07,230 --> 00:12:09,110 At least for isotropic materials. 224 00:12:09,110 --> 00:12:10,640 And then we could have more general 225 00:12:10,640 --> 00:12:12,950 with anisotropic materials. 226 00:12:12,950 --> 00:12:14,290 That would be fun. 227 00:12:14,290 --> 00:12:16,190 All of those things are fun. 228 00:12:16,190 --> 00:12:19,280 But Laplace's equation is the most fun of all. 229 00:12:19,280 --> 00:12:23,280 So let me take f to be zero, c to be one, 230 00:12:23,280 --> 00:12:26,050 get back up to Laplace's equation, 231 00:12:26,050 --> 00:12:30,780 and begin to make connections. 232 00:12:30,780 --> 00:12:32,200 Begin to make connections. 233 00:12:32,200 --> 00:12:34,590 OK. 234 00:12:34,590 --> 00:12:39,750 The beautiful connection is the one right here. 235 00:12:39,750 --> 00:12:44,640 When v and u are the same, when v and w are the same, 236 00:12:44,640 --> 00:12:47,370 then I just read off. 237 00:12:47,370 --> 00:12:52,730 If v is the same as w, then how is the potential function, 238 00:12:52,730 --> 00:12:55,810 which is on the left side, connected to the strain 239 00:12:55,810 --> 00:12:59,690 function, which is coming from, any time div w is zero, 240 00:12:59,690 --> 00:13:02,660 I've got a stream function in 2-D, here. 241 00:13:02,660 --> 00:13:06,130 The connection is just, those two match. 242 00:13:06,130 --> 00:13:14,890 du/dx is the same as ds/dy, and du/dy 243 00:13:14,890 --> 00:13:17,750 is the same as minus ds/dx. 244 00:13:17,750 --> 00:13:22,970 You see that the two sides of our world are coming together. 245 00:13:22,970 --> 00:13:28,080 We're dealing with flows that are both gradient flows, 246 00:13:28,080 --> 00:13:33,170 they come from a potential, you could often 247 00:13:33,170 --> 00:13:38,000 say potential flows, I see the word ideal flows coming in. 248 00:13:38,000 --> 00:13:41,300 These are very special flows. 249 00:13:41,300 --> 00:13:47,770 So aero couldn't work entirely with these flows, of course. 250 00:13:47,770 --> 00:13:50,700 These are such ideal flows. 251 00:13:50,700 --> 00:13:52,760 Proper aerodynamics, you've got shocks, 252 00:13:52,760 --> 00:13:54,650 you've got all sorts of stuff going on. 253 00:13:54,650 --> 00:13:58,810 But in a region where everything's beautiful, 254 00:13:58,810 --> 00:14:04,430 then you get back to this, total steady state, steady flow, 255 00:14:04,430 --> 00:14:06,340 steady potential flow. 256 00:14:06,340 --> 00:14:10,240 We've got those equations that connect u 257 00:14:10,240 --> 00:14:14,130 on one side with the divergence business, 258 00:14:14,130 --> 00:14:16,330 the stream function on the other side. 259 00:14:16,330 --> 00:14:19,710 So I want to focus on these. 260 00:14:19,710 --> 00:14:22,700 Actually, I mean, so the heart of Laplace's equation 261 00:14:22,700 --> 00:14:24,680 is in there. 262 00:14:24,680 --> 00:14:26,280 OK. 263 00:14:26,280 --> 00:14:30,522 First, do you know the names of those two guys? 264 00:14:30,522 --> 00:14:31,980 I shouldn't call them guys, they're 265 00:14:31,980 --> 00:14:35,820 the greatest mathematicians ever. 266 00:14:35,820 --> 00:14:38,530 Maybe after Gauss. 267 00:14:38,530 --> 00:14:43,020 Do you know whose names are associated with those two 268 00:14:43,020 --> 00:14:45,530 equations? 269 00:14:45,530 --> 00:14:47,140 Well, Lagrange was great. 270 00:14:47,140 --> 00:14:49,300 I'm not saying anything about Lagrange. 271 00:14:49,300 --> 00:14:51,220 But he didn't do this. 272 00:14:51,220 --> 00:14:54,910 So two, one French and one German. 273 00:14:54,910 --> 00:15:00,070 So the French guy's name is Cauchy. 274 00:15:00,070 --> 00:15:04,260 And the German is -- so these are Cauchy, 275 00:15:04,260 --> 00:15:07,770 and the German is a really fantastic guy. 276 00:15:07,770 --> 00:15:10,520 Anybody know his name? 277 00:15:10,520 --> 00:15:11,450 Cauchy-Riemann. 278 00:15:11,450 --> 00:15:11,950 Yeah. 279 00:15:11,950 --> 00:15:15,320 The other name is Riemann. 280 00:15:15,320 --> 00:15:21,170 Cauchy-Riemann equations, that connect the two pieces, 281 00:15:21,170 --> 00:15:22,180 u and s. 282 00:15:22,180 --> 00:15:24,660 And they're the subject of an enormous theory 283 00:15:24,660 --> 00:15:27,110 that we'll just touch on here. 284 00:15:27,110 --> 00:15:27,820 OK. 285 00:15:27,820 --> 00:15:32,140 And we'll see them graphically, and we'll find solutions. 286 00:15:32,140 --> 00:15:35,720 So this is one way to pose our problem. 287 00:15:35,720 --> 00:15:39,890 Notice something here. 288 00:15:39,890 --> 00:15:42,250 I think that if we have these equations, 289 00:15:42,250 --> 00:15:45,130 a solution u should satisfy Laplace's equation. 290 00:15:45,130 --> 00:15:49,460 Because this equality will take us around the loop. 291 00:15:49,460 --> 00:15:54,240 So do you see that if I could solve these two -- 292 00:15:54,240 --> 00:15:55,740 here's the point. 293 00:15:55,740 --> 00:15:59,470 I'm going to get Laplace's equation solved by u. 294 00:15:59,470 --> 00:16:03,630 Laplace's equation will also be solved by s. 295 00:16:03,630 --> 00:16:04,540 The stream function. 296 00:16:04,540 --> 00:16:09,160 So what's going to happen is, I get two solutions. 297 00:16:09,160 --> 00:16:11,100 A pair of solutions. 298 00:16:11,100 --> 00:16:17,870 Laplace's equation, solutions to that come in pairs, u and s. 299 00:16:17,870 --> 00:16:19,800 So we get them two at a time. 300 00:16:19,800 --> 00:16:23,680 And they're connected in this remarkable way. 301 00:16:23,680 --> 00:16:26,470 Let's see, can you see that u will satisfy -- 302 00:16:26,470 --> 00:16:28,140 this is the key to everything. 303 00:16:28,140 --> 00:16:30,681 Does u satisfy Laplace's equation? 304 00:16:30,681 --> 00:16:31,180 Sure. 305 00:16:31,180 --> 00:16:33,660 I take the x derivative of that, and I add 306 00:16:33,660 --> 00:16:35,760 to the y derivative of that. 307 00:16:35,760 --> 00:16:36,990 Do you see, it works. 308 00:16:36,990 --> 00:16:40,450 The x derivative of this, plus the y derivative of this 309 00:16:40,450 --> 00:16:43,890 is exactly the cancellation of the cross that I wanted. 310 00:16:43,890 --> 00:16:51,860 So I do these, I combine those into Laplace. 311 00:16:51,860 --> 00:16:56,760 Shall I just go through that verbally again? 312 00:16:56,760 --> 00:16:59,280 I take the x derivative of this, which gives me 313 00:16:59,280 --> 00:17:01,620 the cross derivative of s. 314 00:17:01,620 --> 00:17:04,260 This asked me to take the y derivative, 315 00:17:04,260 --> 00:17:06,150 so I get the cross derivative again. 316 00:17:06,150 --> 00:17:08,240 With the minus sign, they add to zero. 317 00:17:08,240 --> 00:17:13,530 I just want to point out that also, s 318 00:17:13,530 --> 00:17:15,690 satisfies Laplace's equation. 319 00:17:15,690 --> 00:17:20,960 Can we do that one? 320 00:17:20,960 --> 00:17:24,090 I claim that also, the stream function 321 00:17:24,090 --> 00:17:25,510 solves Laplace's equation. 322 00:17:25,510 --> 00:17:28,940 Because we want the x derivative of ds/dx. 323 00:17:28,940 --> 00:17:31,470 So ds/dx is minus this. 324 00:17:31,470 --> 00:17:32,790 Do you see what's happening? 325 00:17:32,790 --> 00:17:35,480 When I take the x derivative of ds/dx, 326 00:17:35,480 --> 00:17:41,200 I get minus the cross derivative of u for this guy. 327 00:17:41,200 --> 00:17:43,430 It's the y derivative of this, which is 328 00:17:43,430 --> 00:17:45,540 plus the cross derivative of u. 329 00:17:45,540 --> 00:17:46,890 They cancel. 330 00:17:46,890 --> 00:17:50,350 So u and s are just together. 331 00:17:50,350 --> 00:17:53,350 Oh, let's find some solutions. 332 00:17:53,350 --> 00:17:55,900 They're great to find. 333 00:17:55,900 --> 00:17:57,770 And then draw them. 334 00:17:57,770 --> 00:18:03,290 OK, can I find some solutions to Laplace's equation. 335 00:18:03,290 --> 00:18:05,900 And I'm going to find them in pairs. 336 00:18:05,900 --> 00:18:10,970 So I'm going to have a list of u's and their corresponding 337 00:18:10,970 --> 00:18:15,840 s's. 338 00:18:15,840 --> 00:18:18,110 OK. 339 00:18:18,110 --> 00:18:20,210 These are solutions to Laplace. 340 00:18:20,210 --> 00:18:23,400 These are, solve Laplace's equation. 341 00:18:23,400 --> 00:18:26,000 So we've got Laplace's equation in our minds. 342 00:18:26,000 --> 00:18:29,599 Actually, furthermore, they solve Cauchy-Riemann. 343 00:18:29,599 --> 00:18:31,140 Because they're going to be connected 344 00:18:31,140 --> 00:18:34,120 by our Cauchy-Riemann equation. 345 00:18:34,120 --> 00:18:39,130 So, solve Laplace and Cauchy-Riemann. 346 00:18:39,130 --> 00:18:40,120 OK. 347 00:18:40,120 --> 00:18:41,650 Suppose I take u(x,y)=x. 348 00:18:41,650 --> 00:18:44,820 349 00:18:44,820 --> 00:18:48,290 I'm going to start with an easy solution. 350 00:18:48,290 --> 00:18:50,320 That certainly solves Laplace's equation. 351 00:18:50,320 --> 00:18:53,080 You've got Laplace's equation in mind? 352 00:18:53,080 --> 00:18:55,611 Let me write it up here again. 353 00:18:55,611 --> 00:18:56,110 u_xx+u_yy=0. 354 00:18:56,110 --> 00:18:59,060 355 00:18:59,060 --> 00:19:02,030 I take the chance to write it again, 356 00:19:02,030 --> 00:19:05,340 to do it in this little bit shorter notation, 357 00:19:05,340 --> 00:19:08,680 just subscripts instead of partials. 358 00:19:08,680 --> 00:19:10,800 And also, s_xx+s_yy=0. 359 00:19:10,800 --> 00:19:13,730 360 00:19:13,730 --> 00:19:18,980 But most of all, the Cauchy-Riemann 361 00:19:18,980 --> 00:19:21,220 that connects the two. 362 00:19:21,220 --> 00:19:25,880 Well, does u=x solve Laplace's equation? 363 00:19:25,880 --> 00:19:27,490 Of course it does. 364 00:19:27,490 --> 00:19:31,640 The second x derivative, if the function is x, is zero. 365 00:19:31,640 --> 00:19:34,540 And the second y derivative is very, very zero. 366 00:19:34,540 --> 00:19:37,400 [LAUGHTER] So what's s? 367 00:19:37,400 --> 00:19:40,260 What's the s that goes with it? 368 00:19:40,260 --> 00:19:43,400 It'll be simple, too. 369 00:19:43,400 --> 00:19:47,220 So the s -- so u is x, right? 370 00:19:47,220 --> 00:19:49,670 I'm starting with this x. 371 00:19:49,670 --> 00:19:52,160 So du/dx is one. 372 00:19:52,160 --> 00:19:55,570 So what do you figure s is? 373 00:19:55,570 --> 00:20:00,670 If du/dx -- see, u is just x itself, so du/dx is only a one. 374 00:20:00,670 --> 00:20:03,180 That derivative was easy. 375 00:20:03,180 --> 00:20:06,620 Then ds/dy is supposed to be one, 376 00:20:06,620 --> 00:20:09,590 and ds/dx is supposed to be zero, I guess. 377 00:20:09,590 --> 00:20:13,380 Do you see what s is? y. 378 00:20:13,380 --> 00:20:16,750 S is y. s is y. 379 00:20:16,750 --> 00:20:19,310 Of course, that solves Laplace's equation, too, 380 00:20:19,310 --> 00:20:20,890 and it solves Cauchy-Riemann. 381 00:20:20,890 --> 00:20:24,030 The x derivative of this is the y derivative of that. 382 00:20:24,030 --> 00:20:25,390 One equal one. 383 00:20:25,390 --> 00:20:28,550 And the y derivative of that is minus the x derivative 384 00:20:28,550 --> 00:20:30,480 of that, zero equal zero. 385 00:20:30,480 --> 00:20:33,350 OK, so that's an easy one. 386 00:20:33,350 --> 00:20:35,620 I'm going to go up a level. 387 00:20:35,620 --> 00:20:37,870 I want to take a second degree. 388 00:20:37,870 --> 00:20:40,890 So my next guy in the list will be -- a pair, 389 00:20:40,890 --> 00:20:48,210 it's a list of pairs -- will be x squared minus y squared. 390 00:20:48,210 --> 00:20:50,500 First of all, it better not be in that list 391 00:20:50,500 --> 00:20:53,320 unless it solves Laplace's equation. 392 00:20:53,320 --> 00:20:56,080 And then if it is, we'll find an s. 393 00:20:56,080 --> 00:20:59,700 So plug it in mentally, can you plug this 394 00:20:59,700 --> 00:21:01,180 into Laplace's equation? 395 00:21:01,180 --> 00:21:05,200 What's the second x derivative of this function? 396 00:21:05,200 --> 00:21:06,720 Two. 397 00:21:06,720 --> 00:21:10,300 Right? xx brings down a two. 398 00:21:10,300 --> 00:21:13,300 What's the second y derivative? 399 00:21:13,300 --> 00:21:15,270 Minus two, from this term. 400 00:21:15,270 --> 00:21:19,111 And then put them into Laplace's equation, two minus two. 401 00:21:19,111 --> 00:21:19,610 Correct. 402 00:21:19,610 --> 00:21:20,430 Zero. 403 00:21:20,430 --> 00:21:21,560 All right. 404 00:21:21,560 --> 00:21:24,070 Now I'm looking for the s that goes with it. 405 00:21:24,070 --> 00:21:26,160 The other one in the pair. 406 00:21:26,160 --> 00:21:29,490 OK, maybe I'd better think through what -- 407 00:21:29,490 --> 00:21:37,180 so this is supposed to give me the s, du/dx -- 408 00:21:37,180 --> 00:21:45,130 let me copy Cauchy-Riemann here, so we can just focus entirely 409 00:21:45,130 --> 00:21:46,440 on that board. 410 00:21:46,440 --> 00:21:54,780 So du/dx, this is 2x in my example. du/dy is minus 2y. 411 00:21:54,780 --> 00:21:59,370 So what am I learning? ds/dy should be 2x. 412 00:21:59,370 --> 00:22:02,500 ds/dx should be 2y. 413 00:22:02,500 --> 00:22:04,600 There are minus signs on both there. 414 00:22:04,600 --> 00:22:05,180 What's s? 415 00:22:05,180 --> 00:22:08,220 Do you see s? 416 00:22:08,220 --> 00:22:13,660 The y derivative is 2x, the x derivative is 2y, 417 00:22:13,660 --> 00:22:21,851 and that stream function is 2xy. 418 00:22:21,851 --> 00:22:22,350 Right? 419 00:22:22,350 --> 00:22:24,650 2xy. 420 00:22:24,650 --> 00:22:28,100 Because the x derivative of this is 2y, 421 00:22:28,100 --> 00:22:31,230 and the y derivative of this is 2x. 422 00:22:31,230 --> 00:22:34,520 And we saw 2xy last time also. 423 00:22:34,520 --> 00:22:38,940 And of course, it solves Laplace's equation easily. 424 00:22:38,940 --> 00:22:43,040 Plug that in, the second derivative is zero. 425 00:22:43,040 --> 00:22:45,640 The second x derivative is zero, second y derivative 426 00:22:45,640 --> 00:22:47,630 is zero, everything. 427 00:22:47,630 --> 00:22:50,480 So that's a pair. 428 00:22:50,480 --> 00:22:55,560 This is a nice pair. 429 00:22:55,560 --> 00:22:59,240 You want to shoot for third degree? 430 00:22:59,240 --> 00:23:04,510 We could maybe figure out third degree, just by jiggling it. 431 00:23:04,510 --> 00:23:06,950 After that, we're going to need an idea 432 00:23:06,950 --> 00:23:08,860 to get up to fourth degree. 433 00:23:08,860 --> 00:23:11,310 Let me try third degree. 434 00:23:11,310 --> 00:23:13,100 Cubics, now. 435 00:23:13,100 --> 00:23:16,470 So I'm looking -- first I just want to get somebody here. 436 00:23:16,470 --> 00:23:19,260 So it's some x cubed. 437 00:23:19,260 --> 00:23:22,110 And then I'm going to need some more stuff, because x cubed 438 00:23:22,110 --> 00:23:26,360 by itself certainly won't work. 439 00:23:26,360 --> 00:23:28,260 I need something more. 440 00:23:28,260 --> 00:23:30,820 And I want to plug it into Laplace's equation 441 00:23:30,820 --> 00:23:33,670 and figure out what should it be? 442 00:23:33,670 --> 00:23:37,590 OK, so when I plug this into Laplace's equation, 443 00:23:37,590 --> 00:23:39,710 what do I get? 444 00:23:39,710 --> 00:23:44,240 Let me do Laplace's equation over here, 445 00:23:44,240 --> 00:23:51,270 to try to get the u of degree three. 446 00:23:51,270 --> 00:23:56,260 So what's u_xx, so far? 447 00:23:56,260 --> 00:23:57,930 6x, right? 448 00:23:57,930 --> 00:24:01,040 Bring down, we've got 3x squared, then we get 6x, 449 00:24:01,040 --> 00:24:03,350 so I've got a 6x. 450 00:24:03,350 --> 00:24:09,010 And I'm looking for -- so u_yy should be minus 6x to cancel 451 00:24:09,010 --> 00:24:10,160 that. 452 00:24:10,160 --> 00:24:13,530 So what do I want there? 453 00:24:13,530 --> 00:24:15,190 What do I need? 454 00:24:15,190 --> 00:24:19,420 I need a minus, I'm sure of that. 455 00:24:19,420 --> 00:24:25,370 So the second y derivative should be this 6x deal. 456 00:24:25,370 --> 00:24:29,110 What do I want? 457 00:24:29,110 --> 00:24:30,550 3xy squared? 458 00:24:30,550 --> 00:24:31,800 That sounds good. 459 00:24:31,800 --> 00:24:34,250 Let me write it down and see if it is good. 460 00:24:34,250 --> 00:24:34,940 OK. 461 00:24:34,940 --> 00:24:37,230 The second y derivative. 462 00:24:37,230 --> 00:24:38,140 Yes. 463 00:24:38,140 --> 00:24:41,730 We'll bring down a two, and then the y's will disappear 464 00:24:41,730 --> 00:24:43,560 and I'll have the minus 6x. 465 00:24:43,560 --> 00:24:44,430 Golden. 466 00:24:44,430 --> 00:24:46,380 OK, that's great. 467 00:24:46,380 --> 00:24:48,710 That's great. 468 00:24:48,710 --> 00:24:49,820 Is that correct? 469 00:24:49,820 --> 00:24:51,500 I mean, it's great, but is it right? 470 00:24:51,500 --> 00:24:52,350 Yes. 471 00:24:52,350 --> 00:24:52,990 Yes. 472 00:24:52,990 --> 00:24:54,100 OK. 473 00:24:54,100 --> 00:25:01,020 Now, you have faith that there's another one? 474 00:25:01,020 --> 00:25:04,260 Well, yeah, there is another one. 475 00:25:04,260 --> 00:25:06,570 Cauchy-Riemann never let us down. 476 00:25:06,570 --> 00:25:09,630 There will be an s that'll go with that, 477 00:25:09,630 --> 00:25:12,120 that'll solve a Cauchy-Riemann equation, 478 00:25:12,120 --> 00:25:13,360 and it'll look like it. 479 00:25:13,360 --> 00:25:21,820 And I think, I think -- and I just sort of reverse x and y 480 00:25:21,820 --> 00:25:24,900 to get another one, because if I exchange x and y, 481 00:25:24,900 --> 00:25:27,470 I'm still OK with Laplace's equation. 482 00:25:27,470 --> 00:25:35,050 I think something like 3yx squared minus y cubed. 483 00:25:35,050 --> 00:25:39,000 If that worked, then this one should work, too. 484 00:25:39,000 --> 00:25:44,310 Because I just switched x and y, and I think 485 00:25:44,310 --> 00:25:45,480 it'll work right here. 486 00:25:45,480 --> 00:25:48,450 The second x derivative will be 6y, 487 00:25:48,450 --> 00:25:52,560 and the second y derivative will be minus 6y. 488 00:25:52,560 --> 00:25:54,490 I think that's good. 489 00:25:54,490 --> 00:25:58,720 OK. 490 00:25:58,720 --> 00:26:01,470 Let's put this list on hold for a moment. 491 00:26:01,470 --> 00:26:03,990 Something's -- obviously, there's some pattern here that 492 00:26:03,990 --> 00:26:06,380 we've got to locate. 493 00:26:06,380 --> 00:26:08,230 Can I put it on hold for a moment 494 00:26:08,230 --> 00:26:11,990 and take, for example, the graph of this one. 495 00:26:11,990 --> 00:26:15,140 And I want to draw the pictures. 496 00:26:15,140 --> 00:26:19,020 Before I get a complete list, I want 497 00:26:19,020 --> 00:26:22,470 to draw the pictures of these functions. 498 00:26:22,470 --> 00:26:24,300 And what do I mean by pictures? 499 00:26:24,300 --> 00:26:31,970 I mean draw the -- so now I'm taking the u to be x squared 500 00:26:31,970 --> 00:26:36,300 minus y squared, and the s to be 2xy. 501 00:26:36,300 --> 00:26:41,720 And I want to draw those, I want to draw the vector field 502 00:26:41,720 --> 00:26:43,930 of the gradients of those guys. 503 00:26:43,930 --> 00:26:47,820 OK, so this is the u -- I want to draw -- 504 00:26:47,820 --> 00:26:51,300 this is the potential x squared minus y squared. 505 00:26:51,300 --> 00:26:54,820 So what are they equipotential curves? 506 00:26:54,820 --> 00:26:58,010 So this is now -- I'm drawing the flow. 507 00:26:58,010 --> 00:27:02,100 And to draw the flow, I draw the curves on which the potential 508 00:27:02,100 --> 00:27:06,300 is a constant. x squared minus y squared equal constant. 509 00:27:06,300 --> 00:27:08,230 So what kind of a curve is x squared 510 00:27:08,230 --> 00:27:10,740 minus y squared equal constant? 511 00:27:10,740 --> 00:27:12,360 It's a hyperbola. 512 00:27:12,360 --> 00:27:15,150 It's a hyperbola. 513 00:27:15,150 --> 00:27:18,430 So x squared minus y squared equal, let's take one, 514 00:27:18,430 --> 00:27:21,080 for example. 515 00:27:21,080 --> 00:27:22,910 As one constant. 516 00:27:22,910 --> 00:27:27,940 So x=1 will be on the curve, when y is zero. 517 00:27:27,940 --> 00:27:30,640 And then what else will be on the curve? 518 00:27:30,640 --> 00:27:37,620 If x is a little bigger, like two -- so here's (1, 0). 519 00:27:37,620 --> 00:27:42,240 That point is certainly -- x squared minus y squared is one 520 00:27:42,240 --> 00:27:43,030 for that. 521 00:27:43,030 --> 00:27:45,730 Suppose I go out to x=2. 522 00:27:45,730 --> 00:27:50,770 What should y be then, to make this right? 523 00:27:50,770 --> 00:27:52,250 Where's the curve going? 524 00:27:52,250 --> 00:27:56,310 When x is two, what is y? 525 00:27:56,310 --> 00:27:59,450 Square root of three, plus or minus. 526 00:27:59,450 --> 00:28:01,930 So square root of three is something like this, 527 00:28:01,930 --> 00:28:04,510 up or down. 528 00:28:04,510 --> 00:28:09,050 It's a curve, like so. 529 00:28:09,050 --> 00:28:14,120 It's a hyperbola. 530 00:28:14,120 --> 00:28:20,350 And now, if I change c to four, let's say, 531 00:28:20,350 --> 00:28:26,200 then it'll go through (2, 0), and it'll go up this way. 532 00:28:26,200 --> 00:28:34,700 And if I change x to something very small, it'll still -- oh, 533 00:28:34,700 --> 00:28:38,880 there's a Greek word, asymptotes. 534 00:28:38,880 --> 00:28:39,380 Oh, yeah. 535 00:28:39,380 --> 00:28:43,190 What do I get if -- the asymptote is when this is zero. 536 00:28:43,190 --> 00:28:43,900 Yeah. 537 00:28:43,900 --> 00:28:47,680 When that is zero, what's my curve? 538 00:28:47,680 --> 00:28:54,100 What are x and y? 539 00:28:54,100 --> 00:28:57,610 They'll be the same. x will be y, or minus y. 540 00:28:57,610 --> 00:29:02,100 I'll be on that straight line, or on this straight line. 541 00:29:02,100 --> 00:29:04,820 So all these other hyperbolas are 542 00:29:04,820 --> 00:29:10,270 kind of asymptotic, whatever the word is. 543 00:29:10,270 --> 00:29:11,600 Right, do you see them? 544 00:29:11,600 --> 00:29:15,310 As a bunch of hyperbolas? 545 00:29:15,310 --> 00:29:23,330 And actually, more hyperbolas -- well, yeah. 546 00:29:23,330 --> 00:29:27,390 Let's see, back when I had a one there, 547 00:29:27,390 --> 00:29:32,580 I took x and y to be positive, and I got that hyperbola. 548 00:29:32,580 --> 00:29:36,150 But since I'm squaring them there, 549 00:29:36,150 --> 00:29:39,470 also these hyperbolas are here. 550 00:29:39,470 --> 00:29:46,650 So these same guys are on that side of the picture. 551 00:29:46,650 --> 00:29:49,850 Those are the equipotentials. 552 00:29:49,850 --> 00:29:55,200 So these are the equipotentials. 553 00:29:55,200 --> 00:29:57,240 OK. 554 00:29:57,240 --> 00:30:03,500 Now, let me draw s. 555 00:30:03,500 --> 00:30:08,920 So those will be the equi -- no, I don't want to say equi-stream 556 00:30:08,920 --> 00:30:09,430 functions. 557 00:30:09,430 --> 00:30:12,120 That's awkward. 558 00:30:12,120 --> 00:30:17,740 So now, I want to draw s equal constant, like one or whatever. 559 00:30:17,740 --> 00:30:21,320 So I've drawn this with a whole lot of constants, 560 00:30:21,320 --> 00:30:24,780 and now I want to draw the other guys. 561 00:30:24,780 --> 00:30:27,730 What do those curves look like? 562 00:30:27,730 --> 00:30:30,380 And what's their name? 563 00:30:30,380 --> 00:30:34,170 A curve on which the stream function is a constant 564 00:30:34,170 --> 00:30:37,220 has a nice name, streamline. 565 00:30:37,220 --> 00:30:40,940 So now I'm going to draw the streamlines. 566 00:30:40,940 --> 00:30:45,470 And what are the streamlines? 567 00:30:45,470 --> 00:30:47,640 The streamlines will be the curves 568 00:30:47,640 --> 00:30:50,400 that the actual material flows. 569 00:30:50,400 --> 00:30:56,630 If you drop a leaf into this flow, and you watch it, 570 00:30:56,630 --> 00:31:00,070 it'll flow along a streamline. 571 00:31:00,070 --> 00:31:02,010 And we can draw those lines. 572 00:31:02,010 --> 00:31:03,830 So what are those? 573 00:31:03,830 --> 00:31:09,060 2xy=1, or the equation y=1/(2x). 574 00:31:09,060 --> 00:31:11,200 That's also a hyperbola, right? 575 00:31:11,200 --> 00:31:12,510 That's also a hyperbola. 576 00:31:12,510 --> 00:31:18,260 This is a fantastic picture, in which 577 00:31:18,260 --> 00:31:21,820 we have two sets of hyperbolas. 578 00:31:21,820 --> 00:31:25,520 We're second degree, that's why we're getting two hyperbolas. 579 00:31:25,520 --> 00:31:29,650 I'm not going to tackle drawing -- MATLAB could do it -- 580 00:31:29,650 --> 00:31:33,470 drawing the equipotentials and the streamlines for this guy. 581 00:31:33,470 --> 00:31:37,570 Oh, but I'm willing to tackle this one. 582 00:31:37,570 --> 00:31:42,980 What are the equipotentials and the streamlines for the easiest 583 00:31:42,980 --> 00:31:46,260 one in the list, there? 584 00:31:46,260 --> 00:31:47,530 Can I just draw that one? 585 00:31:47,530 --> 00:31:51,170 Because it makes a point very clear, that we'll see when 586 00:31:51,170 --> 00:31:53,100 we draw these. 587 00:31:53,100 --> 00:31:59,170 Okay, so what's the picture, the corresponding picture? 588 00:31:59,170 --> 00:32:01,500 Here is the xy-plane again. 589 00:32:01,500 --> 00:32:07,930 What are the equipotentials for this pair? 590 00:32:07,930 --> 00:32:09,960 And the streamlines. 591 00:32:09,960 --> 00:32:15,340 The equipotentials are x equal constant, what are those? 592 00:32:15,340 --> 00:32:17,680 Those are lines, vertical lines. 593 00:32:17,680 --> 00:32:21,120 So the equipotentials are just vertical lines. 594 00:32:21,120 --> 00:32:26,720 Equipotentials, x equal a constant. 595 00:32:26,720 --> 00:32:30,130 And what are the streamlines? 596 00:32:30,130 --> 00:32:32,450 Horizontal lines. 597 00:32:32,450 --> 00:32:34,240 Streamlines go this way. 598 00:32:34,240 --> 00:32:37,110 And what's the great point about these? 599 00:32:37,110 --> 00:32:40,610 These are the streamlines, s equal constant. 600 00:32:40,610 --> 00:32:45,730 And of course, what do you notice here? 601 00:32:45,730 --> 00:32:47,870 They're perpendicular. 602 00:32:47,870 --> 00:32:50,070 The streamlines are perpendicular 603 00:32:50,070 --> 00:32:52,400 to the equipotentials. 604 00:32:52,400 --> 00:32:53,520 And why? 605 00:32:53,520 --> 00:32:57,780 It's because -- you remember we talked about, 606 00:32:57,780 --> 00:32:59,580 what does the gradient mean? 607 00:32:59,580 --> 00:33:01,320 Which way does the gradient point? 608 00:33:01,320 --> 00:33:04,930 It points perpendicular to these equipotentials. 609 00:33:04,930 --> 00:33:09,590 And in this case, all these equipotentials are parallel, 610 00:33:09,590 --> 00:33:12,780 and the perpendicular lines are the streamlines, 611 00:33:12,780 --> 00:33:15,000 and they're all parallel. 612 00:33:15,000 --> 00:33:19,480 Now over here, we haven't got straight lines. 613 00:33:19,480 --> 00:33:23,260 but we still have the beautiful figure. 614 00:33:23,260 --> 00:33:24,470 Now I'm ready to tackle it. 615 00:33:24,470 --> 00:33:26,820 I'll draw these curves. 616 00:33:26,820 --> 00:33:31,690 They're hyperbolas, like y=1/2x. 617 00:33:31,690 --> 00:33:34,560 If I just make that y=1/2x. 618 00:33:34,560 --> 00:33:36,780 That's a line that comes down this way. 619 00:33:36,780 --> 00:33:39,490 Let me try to draw it. 620 00:33:39,490 --> 00:33:41,120 I'll use dashed lines. 621 00:33:41,120 --> 00:33:46,890 Of course, as x gets bigger, y gets smaller. 622 00:33:46,890 --> 00:33:52,990 But y never makes it to zero, because if y was zero, 623 00:33:52,990 --> 00:33:55,550 no x would work. 624 00:33:55,550 --> 00:34:01,550 But if I change that one to to a four, I've got a bigger -- 625 00:34:01,550 --> 00:34:03,900 this is coming out here. 626 00:34:03,900 --> 00:34:09,390 If I change that to something very small, 627 00:34:09,390 --> 00:34:12,230 I'll get one that's coming -- 628 00:34:12,230 --> 00:34:17,200 Do you see how the picture works? 629 00:34:17,200 --> 00:34:19,670 These are all right angles. 630 00:34:19,670 --> 00:34:20,850 That's the great thing. 631 00:34:20,850 --> 00:34:23,240 Right angles. 632 00:34:23,240 --> 00:34:27,950 90 degree angles. 633 00:34:27,950 --> 00:34:31,660 Between the streamlines and the equipotentials. 634 00:34:31,660 --> 00:34:33,740 You may have seen this before, and now 635 00:34:33,740 --> 00:34:36,280 I just want to ask you why. 636 00:34:36,280 --> 00:34:39,920 Why are those 90 degrees? 637 00:34:39,920 --> 00:34:44,920 We kind of see it physically, that the gradient -- 638 00:34:44,920 --> 00:34:47,430 the flow is in the gradient direction. 639 00:34:47,430 --> 00:34:49,750 And we know that gradients are always 640 00:34:49,750 --> 00:34:54,330 perpendicular to equipotential lines. 641 00:34:54,330 --> 00:34:58,990 The gradient of any function is perpendicular to the level 642 00:34:58,990 --> 00:35:00,440 curves. 643 00:35:00,440 --> 00:35:02,230 That's all we're seeing here. 644 00:35:02,230 --> 00:35:08,230 But we're seeing these two fantastic families of curves. 645 00:35:08,230 --> 00:35:11,060 Equipotentials perpendicular to the streamline. 646 00:35:11,060 --> 00:35:12,940 And the reason they're perpendicular 647 00:35:12,940 --> 00:35:15,750 is Cauchy-Riemann. 648 00:35:15,750 --> 00:35:19,130 Cauchy-Riemann is telling us they're perpendicular. 649 00:35:19,130 --> 00:35:24,610 Because the gradient -- yeah, you see that 650 00:35:24,610 --> 00:35:28,460 they're perpendicular, and this may be overkill to try to give 651 00:35:28,460 --> 00:35:32,180 a proof. 652 00:35:32,180 --> 00:35:39,410 The gradient of u -- what I want to say is, 653 00:35:39,410 --> 00:35:43,320 the gradient of u is a 90 degree rotation of the gradient of s. 654 00:35:43,320 --> 00:35:44,900 Let me put it that way. 655 00:35:44,900 --> 00:35:49,920 The gradient of s -- the gradient of u, 656 00:35:49,920 --> 00:35:57,180 rotate 90 degrees, rotate gradient of u by 90 degrees, 657 00:35:57,180 --> 00:36:04,940 pi/2, and you get grad s. 658 00:36:04,940 --> 00:36:07,470 That's what these equations say. 659 00:36:07,470 --> 00:36:14,680 That if I take the gradient of u -- yeah, let me try to do that. 660 00:36:14,680 --> 00:36:16,960 So I take the gradient of u. 661 00:36:16,960 --> 00:36:19,480 That's [du/dx, du/dy]. 662 00:36:19,480 --> 00:36:25,270 So if I have a vector in 2-D, [du/dx, du/dy]. 663 00:36:25,270 --> 00:36:30,500 And I want to rotate a vector by 90 degrees. 664 00:36:30,500 --> 00:36:32,400 What's the result? 665 00:36:32,400 --> 00:36:38,090 Suppose I have a vector [a, b], and I'm looking to rotate it. 666 00:36:38,090 --> 00:36:40,960 What's the vector that goes that way? 667 00:36:40,960 --> 00:36:47,120 If that vector is [a, b], that vector should be what? 668 00:36:47,120 --> 00:36:50,530 You may not have done this, but it's worth just noticing, 669 00:36:50,530 --> 00:36:52,530 and you won't forget it. 670 00:36:52,530 --> 00:36:55,140 It's got to have a zero dot product, right? 671 00:36:55,140 --> 00:36:58,720 So I'm looking for a vector, here. 672 00:36:58,720 --> 00:37:01,450 This went out a and up b. 673 00:37:01,450 --> 00:37:03,130 I'm looking for a vector there that's 674 00:37:03,130 --> 00:37:05,350 perpendicular to this vector. 675 00:37:05,350 --> 00:37:08,260 So what should it do? 676 00:37:08,260 --> 00:37:11,360 What am I going to put there? 677 00:37:11,360 --> 00:37:12,880 A b. 678 00:37:12,880 --> 00:37:16,330 And what am I going to put -- oh no. 679 00:37:16,330 --> 00:37:17,480 It went backwards, sorry. 680 00:37:17,480 --> 00:37:21,540 When I put there, I should have put -- minus b. 681 00:37:21,540 --> 00:37:23,820 And what goes there? a. 682 00:37:23,820 --> 00:37:25,130 Yeah. 683 00:37:25,130 --> 00:37:28,760 That's the perpendicular one. 684 00:37:28,760 --> 00:37:29,920 Right? 685 00:37:29,920 --> 00:37:31,940 That's the 90 degree rotation. 686 00:37:31,940 --> 00:37:34,560 And that's what Cauchy-Riemann is doing for us. 687 00:37:34,560 --> 00:37:38,490 I take the gradient, I take this vector. 688 00:37:38,490 --> 00:37:42,270 Now I rotate by 90 degrees, means I reverse these, 689 00:37:42,270 --> 00:37:45,410 reverse the sign, and I've got the gradient 690 00:37:45,410 --> 00:37:48,400 of S. Or minus the gradient of S. 691 00:37:48,400 --> 00:37:51,480 I won't say whether the rotation is plus 90 692 00:37:51,480 --> 00:37:55,110 degrees or minus 90 degrees. 693 00:37:55,110 --> 00:38:05,250 This can remain an exercise to see it slowly and clearly. 694 00:38:05,250 --> 00:38:07,990 I'm happy if you see it in the picture there. 695 00:38:07,990 --> 00:38:12,740 And of course, you saw it in this picture, here. 696 00:38:12,740 --> 00:38:16,220 So this is a moment, then, to take a little pause. 697 00:38:16,220 --> 00:38:21,310 Because we've got ten minutes for the great event. 698 00:38:21,310 --> 00:38:23,700 We've got the general idea. 699 00:38:23,700 --> 00:38:26,400 The u equal constants and the s equal constant. 700 00:38:26,400 --> 00:38:29,360 These two families of perpendicular curves, 701 00:38:29,360 --> 00:38:34,560 one telling us where level sets for the potential, 702 00:38:34,560 --> 00:38:37,060 the other telling us the direction of the flow. 703 00:38:37,060 --> 00:38:39,680 The flow goes perpendicular to the level sets. 704 00:38:39,680 --> 00:38:43,250 It's just wonderful. 705 00:38:43,250 --> 00:38:50,540 And now I would like to get the pattern that's going on here 706 00:38:50,540 --> 00:38:52,920 and complete that list. 707 00:38:52,920 --> 00:38:57,910 OK, well that pattern, like, comes out of the blue, 708 00:38:57,910 --> 00:39:01,500 I have to admit. 709 00:39:01,500 --> 00:39:04,800 You might sort of recognize it, that something -- here is, 710 00:39:04,800 --> 00:39:06,880 obviously, stuff to the first power. 711 00:39:06,880 --> 00:39:09,260 Here we've squared something to get here. 712 00:39:09,260 --> 00:39:11,370 Here we've cubed something. 713 00:39:11,370 --> 00:39:14,380 You sort of recognize these numbers, 1 3 3 1, 714 00:39:14,380 --> 00:39:23,200 or one, minus three, whatever. 715 00:39:23,200 --> 00:39:25,730 We're taking powers of something. 716 00:39:25,730 --> 00:39:31,180 And that something is what comes out of the blue. 717 00:39:31,180 --> 00:39:35,370 It's this quantity, x+iy. 718 00:39:35,370 --> 00:39:39,360 719 00:39:39,360 --> 00:39:41,230 Complex variables. 720 00:39:41,230 --> 00:39:44,210 Everything here was real until this moment. 721 00:39:44,210 --> 00:39:47,360 And then I'm saying that the complex number 722 00:39:47,360 --> 00:39:52,150 i, the imaginary number i, the square root of minus one. 723 00:39:52,150 --> 00:39:56,790 Which of course, it's not a real number, 724 00:39:56,790 --> 00:39:59,540 but it has the property, whenever we see i squared, 725 00:39:59,540 --> 00:40:00,740 we write minus one. 726 00:40:00,740 --> 00:40:03,230 So then, we know how to deal with it. 727 00:40:03,230 --> 00:40:04,020 OK. 728 00:40:04,020 --> 00:40:04,780 Sort of. 729 00:40:04,780 --> 00:40:09,080 Anyway, so that complex number I'll call z. 730 00:40:09,080 --> 00:40:14,060 And here's what I think. 731 00:40:14,060 --> 00:40:19,350 I think that these two pieces are the real 732 00:40:19,350 --> 00:40:24,330 and the imaginary parts of z. 733 00:40:24,330 --> 00:40:28,070 Now, these two pieces are the real 734 00:40:28,070 --> 00:40:34,350 and the imaginary parts of z squared. 735 00:40:34,350 --> 00:40:38,210 These two pieces will be the real part 736 00:40:38,210 --> 00:40:42,040 and the imaginary part of z cubed. 737 00:40:42,040 --> 00:40:45,360 And if we check that, and then we 738 00:40:45,360 --> 00:40:48,860 begin to see, why should these satisfy Laplace's equation, 739 00:40:48,860 --> 00:40:50,790 we'll have the whole pattern. 740 00:40:50,790 --> 00:40:54,601 It'll just be x+iy, fourth power, fifth power, 741 00:40:54,601 --> 00:40:55,100 sixth power. 742 00:40:55,100 --> 00:41:00,720 We can make a complete, infinite list of pairs of solutions 743 00:41:00,720 --> 00:41:02,620 to Laplace's equation. 744 00:41:02,620 --> 00:41:09,050 So let me just check what I said about the squares first. 745 00:41:09,050 --> 00:41:11,302 How do you do x+iy squared? 746 00:41:11,302 --> 00:41:12,760 Because that's what I believe we're 747 00:41:12,760 --> 00:41:16,540 seeing in the quadratic list. 748 00:41:16,540 --> 00:41:23,800 So I claim that if I take x+iy squared, just do it normally, 749 00:41:23,800 --> 00:41:32,190 I get x squared, and I get 2ixy, and I get i squared y squared. 750 00:41:32,190 --> 00:41:34,830 Right? 751 00:41:34,830 --> 00:41:39,440 I just squared it, following normal algebra rules. 752 00:41:39,440 --> 00:41:41,560 Now what's the real part? 753 00:41:41,560 --> 00:41:46,980 What's the real part of this x+iy squared? x squared, 754 00:41:46,980 --> 00:41:49,140 this is real. 755 00:41:49,140 --> 00:41:52,020 And this is real, because i squared is minus one, 756 00:41:52,020 --> 00:41:55,850 this says minus y squared, that's our guy. 757 00:41:55,850 --> 00:42:01,190 And the imaginary part is 2xy. 758 00:42:01,190 --> 00:42:04,220 The imaginary part is the part that multiplies i. 759 00:42:04,220 --> 00:42:13,400 So by some magic -- next time is the fun of exploring this magic 760 00:42:13,400 --> 00:42:17,570 -- we get solutions to a real equation. 761 00:42:17,570 --> 00:42:20,360 We get two solutions to a real equation, 762 00:42:20,360 --> 00:42:24,500 by working with this complex thing, x+iy, 763 00:42:24,500 --> 00:42:30,240 and taking things like -- now, if I go to x+iy cubed, well, 764 00:42:30,240 --> 00:42:32,340 let me just say it'll work. 765 00:42:32,340 --> 00:42:39,670 We'll have -- it's like, if you cube something, 766 00:42:39,670 --> 00:42:41,920 you're going to see 1 3 3 1. 767 00:42:41,920 --> 00:42:52,320 You'll have an x cubed, and a 3x squared iy, and a 3xiy twice, 768 00:42:52,320 --> 00:42:57,360 and a one of the iy cubed. 769 00:42:57,360 --> 00:43:00,010 And then you look to see what part is real, and you say, 770 00:43:00,010 --> 00:43:01,750 x cubed is real. 771 00:43:01,750 --> 00:43:07,240 And i squared is the minus, so I have minus 3xy squared. 772 00:43:07,240 --> 00:43:08,870 Golden. 773 00:43:08,870 --> 00:43:10,930 And you look for what part is imaginary, 774 00:43:10,930 --> 00:43:15,090 and you see the 3x squared y, and you see the i cubed -- 775 00:43:15,090 --> 00:43:19,650 so what's i cubed? -- is minus i. 776 00:43:19,650 --> 00:43:24,650 So that's an imaginary term, with the minus we wanted. 777 00:43:24,650 --> 00:43:26,720 So now we've got the whole thing. 778 00:43:26,720 --> 00:43:30,230 We've got solutions to Laplace's equation, 779 00:43:30,230 --> 00:43:37,650 coming from all the powers. 780 00:43:37,650 --> 00:43:41,350 This is now the moment to celebrate. 781 00:43:41,350 --> 00:43:45,520 Because we've got a giant family of solutions 782 00:43:45,520 --> 00:43:50,160 to Laplace's equation. 783 00:43:50,160 --> 00:43:51,690 We've got the real parts. 784 00:43:51,690 --> 00:43:57,850 So u is the real part of x+iy to any power. 785 00:43:57,850 --> 00:44:06,420 And s will be the imaginary part. 786 00:44:06,420 --> 00:44:10,700 And I claim that, just as it held for n equal one, two, 787 00:44:10,700 --> 00:44:15,360 three, for every n, these will be solutions 788 00:44:15,360 --> 00:44:18,110 to Laplace's equation. 789 00:44:18,110 --> 00:44:21,120 Not only that, they'll be connected by Cauchy-Riemann, 790 00:44:21,120 --> 00:44:31,490 so they'll be the potential and the stream function for a flow. 791 00:44:31,490 --> 00:44:34,640 And we've got lots of them. 792 00:44:34,640 --> 00:44:39,040 And, why don't we get even more, because we 793 00:44:39,040 --> 00:44:41,730 have a linear equation here. 794 00:44:41,730 --> 00:44:44,050 So what are we allowed to do, if we have solutions 795 00:44:44,050 --> 00:44:45,620 to a linear equation, what are we 796 00:44:45,620 --> 00:44:47,230 allowed to do with those solutions 797 00:44:47,230 --> 00:44:49,360 to get more solutions? 798 00:44:49,360 --> 00:44:51,790 Combine them. 799 00:44:51,790 --> 00:44:56,500 I can take any combination of these guys. 800 00:44:56,500 --> 00:44:59,970 I can take "or." "And," "or," I don't know which, 801 00:44:59,970 --> 00:45:04,140 should I put "or," "and?" 802 00:45:04,140 --> 00:45:08,070 The real part of any combination of these. 803 00:45:08,070 --> 00:45:13,160 So I could take a combination of, I can take coefficients 804 00:45:13,160 --> 00:45:22,820 c_k, or c_n, maybe I should say c_n, (x+iy) to the nth. 805 00:45:22,820 --> 00:45:25,800 So if I take a combination of solutions, 806 00:45:25,800 --> 00:45:29,760 with coefficient c_n, I still have solutions. 807 00:45:29,760 --> 00:45:38,410 And what will be the twin solution, or the stream 808 00:45:38,410 --> 00:45:40,670 function that goes with this u? 809 00:45:40,670 --> 00:45:45,850 So this is another u, this is virtually a complete family 810 00:45:45,850 --> 00:45:48,980 of u. 811 00:45:48,980 --> 00:45:51,070 Because we have all these coefficients to choose. 812 00:45:51,070 --> 00:45:52,410 And what will be the s? 813 00:45:52,410 --> 00:45:55,360 So what's the corresponding s, then? 814 00:45:55,360 --> 00:46:02,940 It's the imaginary part of -- what? -- the same thing. 815 00:46:02,940 --> 00:46:10,190 We're just taking that same combination 816 00:46:10,190 --> 00:46:11,840 of the special ones. 817 00:46:11,840 --> 00:46:15,030 So the special ones were (x+iy)^n. 818 00:46:15,030 --> 00:46:17,910 And the combinations are those. 819 00:46:17,910 --> 00:46:19,510 Yeah. 820 00:46:19,510 --> 00:46:21,600 These are fantastic solutions. 821 00:46:21,600 --> 00:46:24,840 And now, since I'm blessed with two more minutes, 822 00:46:24,840 --> 00:46:29,460 I get to make them much easier. 823 00:46:29,460 --> 00:46:33,320 Because already when I got up to x cubed and y cubed, 824 00:46:33,320 --> 00:46:35,150 they're looking messy. 825 00:46:35,150 --> 00:46:37,300 You say, okay, great to have all these solutions, 826 00:46:37,300 --> 00:46:40,560 but how am I going to use them? 827 00:46:40,560 --> 00:46:45,250 They're getting more and more complicated as n increases. 828 00:46:45,250 --> 00:46:49,160 But switch to polar coordinates. 829 00:46:49,160 --> 00:46:54,060 Make the same list in polar coordinates. 830 00:46:54,060 --> 00:46:56,850 So again, I'm just going to list the same guys 831 00:46:56,850 --> 00:47:02,505 in polar coordinates r, theta. 832 00:47:02,505 --> 00:47:03,880 Then you'll see the pattern, then 833 00:47:03,880 --> 00:47:05,660 the pattern really jumps out. 834 00:47:05,660 --> 00:47:10,500 So what is x in polar coordinates? 835 00:47:10,500 --> 00:47:14,620 If I switch from x, y, rectangular coordinates, to r, 836 00:47:14,620 --> 00:47:20,330 theta, polar coordinates, x is r*cos(theta). 837 00:47:20,330 --> 00:47:25,510 And y is r*sin(theta). 838 00:47:25,510 --> 00:47:30,360 Well, so far it doesn't look any easier. x and y look fine. 839 00:47:30,360 --> 00:47:36,290 But let me go to the second one. 840 00:47:36,290 --> 00:47:41,180 So this is now r squared cos squared theta, right? 841 00:47:41,180 --> 00:47:48,000 Minus, this is r squared sine squared theta. 842 00:47:48,000 --> 00:47:51,590 Trigonometry comes in. 843 00:47:51,590 --> 00:47:53,520 I have r squared cos squared theta, 844 00:47:53,520 --> 00:47:58,580 minus r squared sine squared theta, I want to simplify that. 845 00:47:58,580 --> 00:48:02,230 So it's got an r squared, and it's got 846 00:48:02,230 --> 00:48:04,910 a cos squared theta minus sine squared theta. 847 00:48:04,910 --> 00:48:07,950 Anybody remember that? cos(2theta)! 848 00:48:07,950 --> 00:48:12,410 849 00:48:12,410 --> 00:48:14,980 And this one. 850 00:48:14,980 --> 00:48:17,830 Well, what do you think is coming? 851 00:48:17,830 --> 00:48:19,520 What do you think is coming here? 852 00:48:19,520 --> 00:48:22,940 I have 2r*cos(theta) times r*sin(theta). 853 00:48:22,940 --> 00:48:27,250 I have two, I've got an r squared, again, 854 00:48:27,250 --> 00:48:30,460 times 2cos(theta)*sin(theta). 855 00:48:30,460 --> 00:48:33,380 So what's 2cos(theta)*sin(theta)? 856 00:48:33,380 --> 00:48:34,460 sin(2theta). 857 00:48:34,460 --> 00:48:36,790 You get it. sin(2theta). 858 00:48:36,790 --> 00:48:41,670 And you know what's coming next, right? 859 00:48:41,670 --> 00:48:46,160 You know that now, the whole family in polar coordinates -- 860 00:48:46,160 --> 00:48:50,590 what's the nth power? 861 00:48:50,590 --> 00:48:53,020 We can now write down the nth one in this list. 862 00:48:53,020 --> 00:49:04,620 It is r^n -- for the nth pair, r^n times cos(n*theta), 863 00:49:04,620 --> 00:49:09,960 and r^n sin(n*theta). 864 00:49:09,960 --> 00:49:16,530 So those are the twins, the u and the s 865 00:49:16,530 --> 00:49:21,960 that we get if we use polar coordinates. 866 00:49:21,960 --> 00:49:23,640 So that's just terrific. 867 00:49:23,640 --> 00:49:25,280 Now we have a whole lot of solutions 868 00:49:25,280 --> 00:49:27,100 to Laplace's equation. 869 00:49:27,100 --> 00:49:31,710 And we have, don't forget, all combinations of them. 870 00:49:31,710 --> 00:49:33,780 And we're really ready to go. 871 00:49:33,780 --> 00:49:41,220 So Friday we'll be solving Laplace's equation in the cases 872 00:49:41,220 --> 00:49:45,720 that we can do it, by pencil and paper, by chalk. 873 00:49:45,720 --> 00:49:48,990 And then, after that, comes solving Laplace's equation 874 00:49:48,990 --> 00:49:51,627 by finite differences and finite elements. 875 00:49:51,627 --> 00:49:52,127