1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high-quality educational resources for free. 6 00:00:10 --> 00:00:11 To make a donation, or to view additional materials from 7 00:00:11 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:24 PROFESSOR STRANG: So, shall we start, as always, 10 00:00:24 --> 00:00:26 just open for questions. 11 00:00:26 --> 00:00:29 About any topic. 12 00:00:29 --> 00:00:34 I listed again trusses, 1-D finite elements. 13 00:00:34 --> 00:00:39 Grad div curl, and I should have squeezed in x+iy, too. 14 00:00:39 --> 00:00:45 As the magic trick for finding solutions to Laplace's 15 00:00:45 --> 00:00:47 equation in 2-D. 16 00:00:47 --> 00:00:50 So those are all certainly topics that are in this 17 00:00:50 --> 00:00:52 part of the course. 18 00:00:52 --> 00:00:57 We didn't really get to 3-D, I'm sorry about that. 19 00:00:57 --> 00:01:00 Where the curl comes in, maybe I can say a few 20 00:01:00 --> 00:01:01 words about curl today. 21 00:01:01 --> 00:01:03 Anyway, questions. 22 00:01:03 --> 00:01:03 Discussion. 23 00:01:03 --> 00:01:04 Yes, thanks. 24 00:01:04 --> 00:01:10 AUDIENCE: [INAUDIBLE] 25 00:01:10 --> 00:01:12 PROFESSOR STRANG: Ooh, let's see. 26 00:01:12 --> 00:01:14 So A^T A for a truss. 27 00:01:14 --> 00:01:16 That's a good question. 28 00:01:16 --> 00:01:20 Trusses, A^T A. 29 00:01:20 --> 00:01:28 I guess I don't know any magic tricks either, so 1 way is to 30 00:01:28 --> 00:01:35 construct A, or A transpose, and then just multiply. 31 00:01:35 --> 00:01:43 A second way to do it would be by the four by four bar element 32 00:01:43 --> 00:01:47 matrices, so go bar by bar. 33 00:01:47 --> 00:01:54 So four by four bar matrices, four by four. 34 00:01:54 --> 00:02:00 So those are already in the A transpose A form. 35 00:02:00 --> 00:02:08 They're little A element, A bar transpose A, jeez, this isn't 36 00:02:08 --> 00:02:13 a great as it should be. 37 00:02:13 --> 00:02:20 A element, but I don't know if that would be, so and then you 38 00:02:20 --> 00:02:24 pop those into their correct places. 39 00:02:24 --> 00:02:28 I don't think I know any great idea beyond that. 40 00:02:28 --> 00:02:33 I think you should really be ready to construct 41 00:02:33 --> 00:02:35 a matrix A, yeah. 42 00:02:35 --> 00:02:44 For a reasonably small truss, of course. 43 00:02:44 --> 00:02:48 And of course the other part of trusses, the fun part is to 44 00:02:48 --> 00:02:55 be able to recognize solutions to Au=0. 45 00:02:56 --> 00:03:00 Possibly by looking at the truss more than 46 00:03:00 --> 00:03:01 by solving Au=0. 47 00:03:03 --> 00:03:05 Yeah, any particular example? 48 00:03:05 --> 00:03:10 Of a truss that I should look at just to pin this down? 49 00:03:10 --> 00:03:14 Any favorite trusses? 50 00:03:14 --> 00:03:16 There was an exam question, what was it, 51 00:03:16 --> 00:03:19 a complicated truss? 52 00:03:19 --> 00:03:22 Let's just create a truss. 53 00:03:22 --> 00:03:25 And just think about it. 54 00:03:25 --> 00:03:29 Maybe I won't create the whole matrix A. 55 00:03:29 --> 00:03:35 Here's a truss. 56 00:03:35 --> 00:03:37 How's that for a truss? 57 00:03:37 --> 00:03:43 So it's got, and let me put no supports on it. 58 00:03:43 --> 00:03:48 Just there's a truss to think about. 59 00:03:48 --> 00:03:53 Probably we won't get to all the gory details but if 60 00:03:53 --> 00:03:57 you look at that truss, what's the shape of A? 61 00:03:57 --> 00:04:03 Shape of the matrix A. 62 00:04:03 --> 00:04:06 I have a row for every bar. 63 00:04:06 --> 00:04:11 So one, two, three, four, five. 64 00:04:11 --> 00:04:16 And how many columns have I got, how many unknown u's, 65 00:04:16 --> 00:04:18 unknown displacements have I got? 66 00:04:18 --> 00:04:21 Eight, two, four, six, eight. 67 00:04:21 --> 00:04:30 So I would expect Au=0 would probably have how many 68 00:04:30 --> 00:04:32 independent solutions? 69 00:04:32 --> 00:04:36 Three. 70 00:04:36 --> 00:04:41 You don't know the exact rank, that's exactly true. 71 00:04:41 --> 00:04:43 There could be more than three, right. 72 00:04:43 --> 00:04:48 So to really pin it down you'd have to be sure you 73 00:04:48 --> 00:04:51 were right about that. 74 00:04:51 --> 00:04:54 So three, at least. 75 00:04:54 --> 00:04:59 And I guess here you could tell me the three solutions to Au=0. 76 00:05:01 --> 00:05:03 Three rigid motions. 77 00:05:03 --> 00:05:07 I could translate it to the right, I could translate it up, 78 00:05:07 --> 00:05:11 and you would know what the u is, so u translating to the 79 00:05:11 --> 00:05:19 right would be <1, 0, 1, 0, 1, 0, 1, 0>, right? 80 00:05:19 --> 00:05:25 That's horizontal motion all the same, rigid motion. 81 00:05:25 --> 00:05:28 And we should certainly discover that if we 82 00:05:28 --> 00:05:31 created A for this truss that Au was zero. 83 00:05:31 --> 00:05:36 And similarly vertical motion and the third one would be? 84 00:05:36 --> 00:05:37 Rotation, rotation. 85 00:05:37 --> 00:05:39 Yeah. 86 00:05:39 --> 00:05:44 So if I did the rotation around there, for example, this 87 00:05:44 --> 00:05:49 guy would, this u would also be a here. 88 00:05:49 --> 00:05:50 This wouldn't move. 89 00:05:50 --> 00:05:55 So I'm putting in the four pieces that would go into u. 90 00:05:55 --> 00:06:00 This one, what would be the u for this, the displacement of 91 00:06:00 --> 00:06:05 that corner of the truss? 92 00:06:05 --> 00:06:06 In a rotation? 93 00:06:06 --> 00:06:12 So my rotation is just swing this whole thing around. 94 00:06:12 --> 00:06:15 Zero. , I think. 95 00:06:15 --> 00:06:16 Right. 96 00:06:16 --> 00:06:19 Because it's not going to go out, it's going to go 97 00:06:19 --> 00:06:21 straight down, . 98 00:06:21 --> 00:06:26 And what do you think this guy is? , let's see. 99 00:06:26 --> 00:06:29 It's going to go this way, so it's going to 100 00:06:29 --> 00:06:31 go forward and down. 101 00:06:31 --> 00:06:32 And I think you're right. 102 00:06:32 --> 00:06:35 One and negative one, yeah. 103 00:06:35 --> 00:06:37 I think that would be right. 104 00:06:37 --> 00:06:46 Yeah, then the truss, we could check each bar. 105 00:06:46 --> 00:06:50 That bar, for example should not change lanes because the 106 00:06:50 --> 00:06:55 movement is perpendicular to the bar and I'm writing ones, 107 00:06:55 --> 00:06:59 but I really should write some much smaller number like 0.1 108 00:06:59 --> 00:07:03 everywhere, or something just so that this isn't 109 00:07:03 --> 00:07:05 a very big angle. 110 00:07:05 --> 00:07:09 And it wouldn't change length to first order. 111 00:07:09 --> 00:07:17 So that that's maybe an example where we see the motions, but 112 00:07:17 --> 00:07:20 we didn't actually create A, and we should be able to create 113 00:07:20 --> 00:07:25 A, don't let me prevent you from thinking about A. 114 00:07:25 --> 00:07:28 Yeah. 115 00:07:28 --> 00:07:29 AUDIENCE: [INAUDIBLE] 116 00:07:29 --> 00:07:32 Should those be ones, or should they be root two over twos? 117 00:07:32 --> 00:07:34 PROFESSOR STRANG: Well, that's a good question and after many 118 00:07:34 --> 00:07:39 years I've figured out that they're ones. 119 00:07:39 --> 00:07:41 But it's a very good question. 120 00:07:41 --> 00:07:45 Let's just see why. 121 00:07:45 --> 00:07:49 Let's look at this bar to be sure it's not stretched. 122 00:07:49 --> 00:07:50 Right? 123 00:07:50 --> 00:07:54 So this guy is moving over by one, and this also 124 00:07:54 --> 00:07:55 by one is my claim. 125 00:07:55 --> 00:07:59 And then this movement down doesn't stretch it to 126 00:07:59 --> 00:08:00 first order, yeah. 127 00:08:00 --> 00:08:04 So I needed to make those guys the same. 128 00:08:04 --> 00:08:07 I guess what I figured out is that if you're rotating around 129 00:08:07 --> 00:08:13 here, then somehow it's the x and y, is that. 130 00:08:13 --> 00:08:14 Anyway. 131 00:08:14 --> 00:08:19 Whatever. 132 00:08:19 --> 00:08:26 So that gives us a chance to do a specific example. 133 00:08:26 --> 00:08:30 OK, but I've dodged the creation of A. 134 00:08:30 --> 00:08:31 Yep, thanks. 135 00:08:31 --> 00:08:37 AUDIENCE: [INAUDIBLE] 136 00:08:37 --> 00:08:41 PROFESSOR STRANG: Sorry, the solution to A transpose A? 137 00:08:41 --> 00:08:46 AUDIENCE: [INAUDIBLE] 138 00:08:46 --> 00:08:47 PROFESSOR STRANG: Yeah, I see, OK. 139 00:08:47 --> 00:08:51 So the reason I stopped here was that A transpose 140 00:08:51 --> 00:08:53 A will be singular. 141 00:08:53 --> 00:08:57 So I wouldn't, like, go ahead to go forward 142 00:08:57 --> 00:08:58 to A transpose Au=F. 143 00:09:00 --> 00:09:08 But if I put on some supports, then of course now all good. 144 00:09:08 --> 00:09:14 So now I have, what's now the shape of A? 145 00:09:14 --> 00:09:17 For this one. 146 00:09:17 --> 00:09:22 I now have this bar is now, forget it, right? 147 00:09:22 --> 00:09:24 This bar is just between two supports. 148 00:09:24 --> 00:09:31 So if we put it in the matrix it'll just be a row of zeroes. 149 00:09:31 --> 00:09:33 Nothing will happen, and we're better off to 150 00:09:33 --> 00:09:34 just knock it out. 151 00:09:34 --> 00:09:39 So I think, now I have, I now have four bars. 152 00:09:39 --> 00:09:41 And how many unknowns? 153 00:09:41 --> 00:09:44 Four, two there and two there. 154 00:09:44 --> 00:09:48 And, do you guess that it's stable? 155 00:09:48 --> 00:09:49 That truss? 156 00:09:49 --> 00:09:51 Yeah, that looks stable to me. 157 00:09:51 --> 00:09:56 So the four by four matrix would be invertable and 158 00:09:56 --> 00:09:58 then I could solve. 159 00:09:58 --> 00:09:59 Good point. 160 00:09:59 --> 00:10:04 Then, A would be four by four, A transpose would be four by 161 00:10:04 --> 00:10:07 four, C would be four by four in between. 162 00:10:07 --> 00:10:11 This is the case that I gave the name for this case. 163 00:10:11 --> 00:10:14 When I have a square matrix, do you remember the name 164 00:10:14 --> 00:10:15 just for the hell of it? 165 00:10:15 --> 00:10:17 Statically determinant. 166 00:10:17 --> 00:10:21 It's determinant because each step determines 167 00:10:21 --> 00:10:22 everything completely. 168 00:10:22 --> 00:10:28 Normally, if I have another bar there, now it would be five by 169 00:10:28 --> 00:10:35 five, and now I really have to do the A transpose C A to get 170 00:10:35 --> 00:10:38 to a four by four invertable. 171 00:10:38 --> 00:10:43 By itself, A would not be invertable. 172 00:10:43 --> 00:10:47 This is the more typical case, where you really have to 173 00:10:47 --> 00:10:49 put all three together. 174 00:10:49 --> 00:10:51 Right. 175 00:10:51 --> 00:10:55 I hope you enjoyed the trusses part, though, and continue 176 00:10:55 --> 00:11:01 to enjoy them this evening. 177 00:11:01 --> 00:11:05 OK, I'll just keep moving to be sure that we 178 00:11:05 --> 00:11:07 cover any other topics. 179 00:11:07 --> 00:11:07 Yeah, thanks. 180 00:11:07 --> 00:11:10 AUDIENCE: [INAUDIBLE] 181 00:11:10 --> 00:11:11 PROFESSOR STRANG: For this matrix? 182 00:11:11 --> 00:11:13 AUDIENCE: [INAUDIBLE] 183 00:11:13 --> 00:11:14 PROFESSOR STRANG: For that truss? 184 00:11:14 --> 00:11:15 Oh my God. 185 00:11:15 --> 00:11:20 OK, let me see. 186 00:11:20 --> 00:11:23 Then can I do one row? 187 00:11:23 --> 00:11:27 OK, of course, you guys are responsible for much more. 188 00:11:27 --> 00:11:29 Alright, which row shall I do? 189 00:11:29 --> 00:11:32 Which bar? 190 00:11:32 --> 00:11:33 A diagonal bar? 191 00:11:33 --> 00:11:38 I knew you'd make it like, you could make up quizzes and I 192 00:11:38 --> 00:11:40 could just sit back here. 193 00:11:40 --> 00:11:43 OK, so let's take this diagonal bar, alright. 194 00:11:43 --> 00:11:46 And we're going to keep that supported. 195 00:11:46 --> 00:11:49 Or not, do you want to keep it supported? 196 00:11:49 --> 00:11:53 OK, so in this case then that end is not moving. 197 00:11:53 --> 00:11:56 So this will be in this, and this bar corresponds 198 00:11:56 --> 00:11:58 to a row of A. 199 00:11:58 --> 00:12:03 And how many non-zeroes will I expect in that row? 200 00:12:03 --> 00:12:04 Just two. 201 00:12:04 --> 00:12:08 Normally four, but I'm not getting any motion down here. 202 00:12:08 --> 00:12:11 So it'll just be two and if that's 45 degrees, shall we 203 00:12:11 --> 00:12:16 say, than that row, I think, would be what? 204 00:12:16 --> 00:12:19 Well, OK. 205 00:12:19 --> 00:12:21 Where do my non-zeroes appear? 206 00:12:21 --> 00:12:26 This is node number one with an H and a V. 207 00:12:26 --> 00:12:30 So I think we have zeroes. 208 00:12:30 --> 00:12:37 If that bar stretches, the connection between displacement 209 00:12:37 --> 00:12:40 and stretching of this bar does not involve this guy. 210 00:12:40 --> 00:12:42 So I think it's zero and zero there. 211 00:12:42 --> 00:12:45 And now, what else is it? 212 00:12:45 --> 00:12:49 So now come the real numbers, which I believe to be cosine 213 00:12:49 --> 00:12:51 and sine of that angle. 214 00:12:51 --> 00:12:56 Because if I, how much does that bar stretch? 215 00:12:56 --> 00:13:01 I know that I'm looking for a cosine and a sine, and if this 216 00:13:01 --> 00:13:04 goes positively, then the bar does stretch. 217 00:13:04 --> 00:13:08 If this goes positively, that does stretch the bar so I'm 218 00:13:08 --> 00:13:13 expecting positive numbers there, like the cosine square 219 00:13:13 --> 00:13:15 root of two over two, and the sine square root 220 00:13:15 --> 00:13:20 of two over two. 221 00:13:20 --> 00:13:25 Well, I dodged the bullet of getting the whole matrix, 222 00:13:25 --> 00:13:27 but maybe that would do it. 223 00:13:27 --> 00:13:30 Why don't we do this the top one? 224 00:13:30 --> 00:13:31 Yeah. 225 00:13:31 --> 00:13:36 Tell me the first, if that's bar one, what would be the 226 00:13:36 --> 00:13:38 first row of the matrix? 227 00:13:38 --> 00:13:42 OK, it involves both of these nodes. 228 00:13:42 --> 00:13:44 But the angle is zero. 229 00:13:44 --> 00:13:48 So that's going to be a little special. 230 00:13:48 --> 00:13:56 So if this goes out horizontally, I should really 231 00:13:56 --> 00:13:57 start with the first one. 232 00:13:57 --> 00:14:00 Is this goes horizontally it compresses the bar, I think 233 00:14:00 --> 00:14:02 we get a minus one there. 234 00:14:02 --> 00:14:05 If it goes vertically, that doesn't do anything. 235 00:14:05 --> 00:14:08 If this goes horizontally it does do something. 236 00:14:08 --> 00:14:11 If it goes vertically it doesn't. 237 00:14:11 --> 00:14:16 I'd say that would be the row of the matrix 238 00:14:16 --> 00:14:22 coming from the top. 239 00:14:22 --> 00:14:24 That would give me the stretching. 240 00:14:24 --> 00:14:26 You remember, I'm always going to, I think of 241 00:14:26 --> 00:14:28 multiplying this by u. 242 00:14:28 --> 00:14:37 I think of multiplying that by u 1 H, u 1 V, u 2 H, u 2 V, and 243 00:14:37 --> 00:14:45 this top row should give me you u 2 H minus u 1 H, which is 244 00:14:45 --> 00:14:57 the stretch in bar one. 245 00:14:57 --> 00:15:03 So that would be a typical one, this would be at least typical 246 00:15:03 --> 00:15:06 of one where I do see a cosine and a sine. 247 00:15:06 --> 00:15:15 And let me just finally add, suppose this was not supported. 248 00:15:15 --> 00:15:17 OK, suppose that's not supported, now I've got 249 00:15:17 --> 00:15:20 a couple more columns to squeeze in. 250 00:15:20 --> 00:15:23 Maybe I can somehow do it here. 251 00:15:23 --> 00:15:25 Can I squeeze in the two more columns? 252 00:15:25 --> 00:15:31 So can you complete the top row of the matrix? 253 00:15:31 --> 00:15:33 Now I've got six columns. 254 00:15:33 --> 00:15:37 Because here's two, here's two, here's two more. 255 00:15:37 --> 00:15:40 What goes on the top row of a matrix now? 256 00:15:40 --> 00:15:44 Zeroes, because this is not affected by bar one. 257 00:15:44 --> 00:15:47 But it is affected by this bar. 258 00:15:47 --> 00:15:49 So it's going to show up in this row, and how 259 00:15:49 --> 00:15:52 will it show up. 260 00:15:52 --> 00:15:53 Two negatives, right. 261 00:15:53 --> 00:15:58 A negative cosine and a negative sine. and at 45 262 00:15:58 --> 00:16:02 degrees I can't tell the difference. 263 00:16:02 --> 00:16:05 Because if these move forward, that compresses the bar. 264 00:16:05 --> 00:16:07 So the minus sign. 265 00:16:07 --> 00:16:15 So again, the rows add up to zero, as we expect when the bar 266 00:16:15 --> 00:16:18 is not touching a support. 267 00:16:18 --> 00:16:26 This is not touching a support, so it adds up to zero. 268 00:16:26 --> 00:16:34 OK, we'll have a truss problem this evening, 269 00:16:34 --> 00:16:39 but not a big messy one. 270 00:16:39 --> 00:16:42 How about finite elements? 271 00:16:42 --> 00:16:45 You guys, do you like finite elements? 272 00:16:45 --> 00:16:49 I'm sort of hoping to make them attractive. 273 00:16:49 --> 00:16:53 I noticed a problem just to give us some specific one to 274 00:16:53 --> 00:16:56 work on, and I don't remember that it was a homework problem. 275 00:16:56 --> 00:17:04 This is Section 3.1, number 18, asks about the equation u''=0. 276 00:17:04 --> 00:17:07 277 00:17:07 --> 00:17:11 Well, we've talked about it in class. 278 00:17:11 --> 00:17:19 With u(0)=0 but u' of slope equals zero at the other end. 279 00:17:19 --> 00:17:23 So what's the picture if I use linear elements? 280 00:17:23 --> 00:17:26 I don't remember how many I used in the problem. 281 00:17:26 --> 00:17:34 Well, it allows you to use n interior guys, one, two, 282 00:17:34 --> 00:17:38 up to n, and then another. 283 00:17:38 --> 00:17:41 This will come in. 284 00:17:41 --> 00:17:45 Or that's the point. 285 00:17:45 --> 00:17:49 OK, so what's the finite element method, it's finite 286 00:17:49 --> 00:17:52 element matrix K for this. 287 00:17:52 --> 00:18:01 So I want to do linear elements and I want to construct K. 288 00:18:01 --> 00:18:03 And, yeah, I guess. 289 00:18:03 --> 00:18:08 Oh, I haven't actually made anything happen 290 00:18:08 --> 00:18:09 to this problem. 291 00:18:09 --> 00:18:14 All zeroes is kind of slow going. u will be the solution, 292 00:18:14 --> 00:18:16 will certainly be zero. 293 00:18:16 --> 00:18:24 So maybe I'd better put in a load here to get some action. 294 00:18:24 --> 00:18:32 OK, well, yeah. 295 00:18:32 --> 00:18:36 So I proposed this question but now, is this a 296 00:18:36 --> 00:18:38 question to think about? 297 00:18:38 --> 00:18:41 I think that's a reasonable example to do. 298 00:18:41 --> 00:18:43 It's got the two types of boundary conditions. 299 00:18:43 --> 00:18:47 It's got the right hand side f, it's got linear elements which 300 00:18:47 --> 00:18:51 means it's kind of doable by hand. 301 00:18:51 --> 00:19:00 And we kind of know what matrix to expect out of it. 302 00:19:00 --> 00:19:05 What matrix do we expect? 303 00:19:05 --> 00:19:09 What do I expect out of linear elements, do you remember the 304 00:19:09 --> 00:19:13 point about linear elements on equally spaced meshes? 305 00:19:13 --> 00:19:17 That just brought back our regular difference matrices. 306 00:19:17 --> 00:19:21 So I'm expecting this stiffness matrix to be 307 00:19:21 --> 00:19:22 a difference matrix. 308 00:19:22 --> 00:19:26 Anyway, the point of this question is, OK, I have a hat 309 00:19:26 --> 00:19:29 function, I have a hat function, I've a hat function, 310 00:19:29 --> 00:19:34 a hat function, and is that the end? 311 00:19:34 --> 00:19:38 Is that the complete list of my trial functions? 312 00:19:38 --> 00:19:40 One more, right? 313 00:19:40 --> 00:19:48 Because this condition is, all my trial and test functions 314 00:19:48 --> 00:19:51 don't have to satisfy this. 315 00:19:51 --> 00:19:55 So I'm allowed, and should have, another guy there. 316 00:19:55 --> 00:19:58 A half hat for that one. 317 00:19:58 --> 00:20:06 You may say, don't let that clown into the finite element 318 00:20:06 --> 00:20:09 space but I think it should be. 319 00:20:09 --> 00:20:13 The solution won't use much of it. 320 00:20:13 --> 00:20:18 Because the solution is going to aim for zero slope. 321 00:20:18 --> 00:20:20 But it's going to need a little, but you see 322 00:20:20 --> 00:20:22 why it needs a little bit, something here? 323 00:20:22 --> 00:20:26 Because this thing has slope down. 324 00:20:26 --> 00:20:29 So if there's some of that in there, there better be 325 00:20:29 --> 00:20:35 somebody else to cancel it. 326 00:20:35 --> 00:20:39 If our approximation is going to have about zero slope. 327 00:20:39 --> 00:20:41 OK, so then. 328 00:20:41 --> 00:20:44 Can you construct a matrix K? 329 00:20:44 --> 00:20:50 Let's see, what's the 2, 3 entry so if I call this number 330 00:20:50 --> 00:20:53 one, this number two, oh, I've already numbered. 331 00:20:53 --> 00:20:57 So number two and number three, so that trial 332 00:20:57 --> 00:20:59 function against that one. 333 00:20:59 --> 00:21:00 What do I? 334 00:21:00 --> 00:21:05 What's my formula for the 2, 3 entry of the stiffness matrix? 335 00:21:05 --> 00:21:08 It's some integral, right? 336 00:21:08 --> 00:21:10 And what do I integrate? 337 00:21:10 --> 00:21:12 I integrate, yeah. 338 00:21:12 --> 00:21:15 And I've got to have to remember. 339 00:21:15 --> 00:21:22 So I do, yeah, my weak I've integrated by parts. 340 00:21:22 --> 00:21:27 So my weak form is the integral of u'*v'*dx equals the 341 00:21:27 --> 00:21:29 integral of f*v*dx. 342 00:21:29 --> 00:21:33 343 00:21:33 --> 00:21:34 That's my weak form. 344 00:21:34 --> 00:21:40 I did two integrations by parts and the integrated 345 00:21:40 --> 00:21:41 term will go away. 346 00:21:41 --> 00:21:43 Because of those zeroes. 347 00:21:43 --> 00:21:52 OK, so K_2,3 will come from this side when I'm using phi_2 348 00:21:52 --> 00:21:57 and phi_3, because I'm taking the phis, the phis and the v's 349 00:21:57 --> 00:22:00 both the same hat function. 350 00:22:00 --> 00:22:08 OK, so what do I get for that? phi_2' is? 351 00:22:08 --> 00:22:12 So this is it, and it overlaps this one. 352 00:22:12 --> 00:22:15 So when it overlaps this phi_2 is coming down 353 00:22:15 --> 00:22:17 and phi_3 is going up. 354 00:22:17 --> 00:22:23 And the slope is 1/h, let's say. 355 00:22:23 --> 00:22:30 So I think I'm integrating us a negative slope, is that right? 356 00:22:30 --> 00:22:32 Times a positive slope. 357 00:22:32 --> 00:22:38 And I'm really only integrating over one h interval. 358 00:22:38 --> 00:22:44 The two overlap only here, where this one's coming down 359 00:22:44 --> 00:22:46 and that one's going up. 360 00:22:46 --> 00:22:51 So I think the x, and the great thing is, of course, 361 00:22:51 --> 00:22:52 we have a constant. 362 00:22:52 --> 00:22:54 So I have minus one over h squared times one over h, 363 00:22:54 --> 00:22:58 I think minus 1 over h. 364 00:22:58 --> 00:22:59 That would be K_2,3. 365 00:23:00 --> 00:23:08 That's a simple example. 366 00:23:08 --> 00:23:19 And then at the end we will see it, we'll see this one, I 367 00:23:19 --> 00:23:21 think we'll get some matrix. 368 00:23:21 --> 00:23:25 We'll have this 1/h outside, I think we'll have something like 369 00:23:25 --> 00:23:29 two minus one, two minus one, minus one, minus one and then 370 00:23:29 --> 00:23:31 only a one from the hat factor. 371 00:23:31 --> 00:23:36 I think it would be that matrix that would be K. 372 00:23:36 --> 00:23:39 I think. 373 00:23:39 --> 00:23:42 Maybe with more, greater size if we have a whole 374 00:23:42 --> 00:23:43 bunch of elements. 375 00:23:43 --> 00:23:47 But that pattern. 376 00:23:47 --> 00:23:49 You're pretty much into this? 377 00:23:49 --> 00:23:54 Yeah, I mean we're doing a lot in this course. 378 00:23:54 --> 00:24:04 I'm really grateful you guys stay with it, and kept to these 379 00:24:04 --> 00:24:07 new ideas, through doing exercises and so on. 380 00:24:07 --> 00:24:11 Because there's a lot here. 381 00:24:11 --> 00:24:14 Well, I thought I'd put an example up, to open up, just 382 00:24:14 --> 00:24:18 to remind you what that language is about there. 383 00:24:18 --> 00:24:23 And to be ready for any question in that topic. 384 00:24:23 --> 00:24:26 Or any question whatever. 385 00:24:26 --> 00:24:31 So I jumped in with finite elements, but I'm ready 386 00:24:31 --> 00:24:37 also to talk about that area of the course. 387 00:24:37 --> 00:24:38 AUDIENCE: [INAUDIBLE] 388 00:24:38 --> 00:24:44 PROFESSOR STRANG: Yeah. x+iy stuff, OK. 389 00:24:44 --> 00:24:47 Basically, any function of x+iy, yeah. 390 00:24:47 --> 00:24:49 Any function. 391 00:24:49 --> 00:24:52 So strictly, yeah, I mean a mathematician would 392 00:24:52 --> 00:24:54 say what, any function? 393 00:24:54 --> 00:25:02 That's, you've open the door to crazy things saying that. 394 00:25:02 --> 00:25:06 So what I really mean is, we have these powers of 395 00:25:06 --> 00:25:09 x+iy, and then we have combinations of them. 396 00:25:09 --> 00:25:13 So the only requirement would be that if I want to take an 397 00:25:13 --> 00:25:16 infinite combination it should, the series should, 398 00:25:16 --> 00:25:19 add up to something. 399 00:25:19 --> 00:25:24 If it has a nice Taylor series then those are the best 400 00:25:24 --> 00:25:25 functions there are. 401 00:25:25 --> 00:25:27 Functions with nice Taylor series'. 402 00:25:27 --> 00:25:29 I'll just say it. 403 00:25:29 --> 00:25:30 Having used those words. 404 00:25:30 --> 00:25:37 Suppose I take that function. 405 00:25:37 --> 00:25:39 There's a function, that's a function z is x+iy. 406 00:25:39 --> 00:25:46 407 00:25:46 --> 00:25:50 But z is shorter to write. 408 00:25:50 --> 00:25:52 So it's not a polynomial, obviously. 409 00:25:52 --> 00:25:54 But it is a function of x+iy. 410 00:25:54 --> 00:26:00 411 00:26:00 --> 00:26:02 Well, tell me this. 412 00:26:02 --> 00:26:05 Where does that function go wrong? 413 00:26:05 --> 00:26:08 So either the z is a function that never goes wrong, 414 00:26:08 --> 00:26:14 right? e^z, that series always converges. 415 00:26:14 --> 00:26:18 Can you tell me the series, if I expand that into a series, 416 00:26:18 --> 00:26:22 what series am I looking at? 417 00:26:22 --> 00:26:28 This is not on the exam, so to speak. 418 00:26:28 --> 00:26:32 Do you know one over one plus something, what's 419 00:26:32 --> 00:26:39 the series for that? 420 00:26:39 --> 00:26:43 Well the constant term, when z is zero is certainly a one. 421 00:26:43 --> 00:26:47 I think the trick, it's this is geometric series, and because 422 00:26:47 --> 00:26:51 it's a z squared it's that. 423 00:26:51 --> 00:26:56 That would be the geometric series. 424 00:26:56 --> 00:26:58 With constant ratio z squared. 425 00:26:58 --> 00:27:03 If I multiply that by that, 1+ z squared times that, 426 00:27:03 --> 00:27:05 everything will cancel and I'll get the one. 427 00:27:05 --> 00:27:08 That's it. 428 00:27:08 --> 00:27:14 So there is a Taylor series for this function. 429 00:27:14 --> 00:27:17 Now, the reason I chose that example is, you could tell 430 00:27:17 --> 00:27:23 me, it doesn't converge if z is too large, right? 431 00:27:23 --> 00:27:28 Is this an analytic function? 432 00:27:28 --> 00:27:30 Where is this a good function and where 433 00:27:30 --> 00:27:33 does it have problems? 434 00:27:33 --> 00:27:38 If z is less than one, and I really mean magnitude of z, 435 00:27:38 --> 00:27:40 so let me draw the z plane. 436 00:27:40 --> 00:27:45 Here's the real part of z that you usually call x, and the 437 00:27:45 --> 00:27:49 imaginary part of z that you usually call y, because z 438 00:27:49 --> 00:27:55 is x+iy, and where will this series converge? 439 00:27:55 --> 00:28:01 It'll converge out as far as this circle. 440 00:28:01 --> 00:28:05 This is the Taylor series around zero. 441 00:28:05 --> 00:28:05 Right? 442 00:28:05 --> 00:28:08 The constant term I found that z=0. 443 00:28:10 --> 00:28:15 Then that series, this function, is great. 444 00:28:15 --> 00:28:20 It's an analytic function, everything, it gives us a 445 00:28:20 --> 00:28:22 solution to Laplace's, this'll be. 446 00:28:22 --> 00:28:26 The real and imaginary parts of that will be the u and the S 447 00:28:26 --> 00:28:28 that solve Laplace's equation. 448 00:28:28 --> 00:28:33 Out to, at least in this circle. 449 00:28:33 --> 00:28:36 But something, there's a problem at the 450 00:28:36 --> 00:28:38 edge of the circle. 451 00:28:38 --> 00:28:42 Now, here's my point. 452 00:28:42 --> 00:28:49 If I think of 1/1+x^2, look at that for a minute. 453 00:28:49 --> 00:28:51 That function has no problems at all, right? 454 00:28:51 --> 00:28:55 1/1+x^2, can let x be anything? 455 00:28:55 --> 00:28:58 With no trouble. 456 00:28:58 --> 00:29:02 But 1/1+z^2, when we look in the complex plane, 457 00:29:02 --> 00:29:04 ah, we find a problem. 458 00:29:04 --> 00:29:07 And where is the problem with this function? 459 00:29:07 --> 00:29:12 At z equals, so everybody's looking at this guy. 460 00:29:12 --> 00:29:14 There's a problem with that function at z=i. 461 00:29:16 --> 00:29:20 And it happens to be not an accident, it's right 462 00:29:20 --> 00:29:22 there on the circle. 463 00:29:22 --> 00:29:26 It's hiding in the complex-- it's not on the real axis. 464 00:29:26 --> 00:29:29 So the real person didn't notice it. 465 00:29:29 --> 00:29:33 But the complex person said ah, that's the problem. 466 00:29:33 --> 00:29:36 There's a singularity there, and of course it's called a 467 00:29:36 --> 00:29:41 pole, and people in so many parts of science are 468 00:29:41 --> 00:29:42 interested in that. 469 00:29:42 --> 00:29:46 Is there any other place that there's a problem? 470 00:29:46 --> 00:29:47 That minus sign. 471 00:29:47 --> 00:29:50 When z is minus i will also get a problem. 472 00:29:50 --> 00:30:00 So this is a function with two poles, those two poles and 473 00:30:00 --> 00:30:05 they're the reason that the series couldn't make it. 474 00:30:05 --> 00:30:07 If going out this way the series doesn't 475 00:30:07 --> 00:30:09 meet any problems. 476 00:30:09 --> 00:30:15 But the series always goes out in a circle, and the first 477 00:30:15 --> 00:30:18 circle, the first guy, the first problem that hits, the 478 00:30:18 --> 00:30:21 series stops converging. 479 00:30:21 --> 00:30:24 By the way, let me ask you a question. 480 00:30:24 --> 00:30:27 Suppose I instead did the Taylor series 481 00:30:27 --> 00:30:29 around this point? 482 00:30:29 --> 00:30:31 Now, what do I mean by that? 483 00:30:31 --> 00:30:33 That's the point one, let's say. 484 00:30:33 --> 00:30:36 What do I mean by that, the Taylor series around one? 485 00:30:36 --> 00:30:40 I'll rewrite the function as one plus, well 486 00:30:40 --> 00:30:41 now, what do I do? 487 00:30:41 --> 00:30:46 I want it in z minus one squared. 488 00:30:46 --> 00:30:47 Oh, gosh. 489 00:30:47 --> 00:30:54 I'm getting beyond what you will care about. 490 00:30:54 --> 00:30:59 Again, if if I expanded, if I wrote the power series 491 00:30:59 --> 00:31:05 in powers of z minus one, what would it work in? 492 00:31:05 --> 00:31:07 And then I'll stop with this example. 493 00:31:07 --> 00:31:12 The circle would reach out until it hit a pole. 494 00:31:12 --> 00:31:14 And it can't make it past that pole. 495 00:31:14 --> 00:31:18 So it would be a circle of radius square root of two, 496 00:31:18 --> 00:31:20 there would be a circle there. 497 00:31:20 --> 00:31:24 If we were going to discuss, and this is really 498 00:31:24 --> 00:31:26 Chapter 5 of the book. 499 00:31:26 --> 00:31:34 I mention it, because you've got a book that explains this. 500 00:31:34 --> 00:31:37 If I thought the center of the universe was there, and then 501 00:31:37 --> 00:31:40 the poles are still here, the circle will make it 502 00:31:40 --> 00:31:42 out to those poles. 503 00:31:42 --> 00:31:46 So I can do Taylor series, I can sort of hook together 504 00:31:46 --> 00:31:49 Taylor series all over the place. 505 00:31:49 --> 00:31:53 And they'll all quit when they reach a pole, but when I put 506 00:31:53 --> 00:31:56 all those circles together I can get all the 507 00:31:56 --> 00:31:57 rest of the plane. 508 00:31:57 --> 00:32:02 OK, so that's something about, I don't know how I got 509 00:32:02 --> 00:32:06 onto that department, but it's amazing. 510 00:32:06 --> 00:32:09 This, so the real and imaginary parts of that would be a 511 00:32:09 --> 00:32:12 flow, would give me a flow. 512 00:32:12 --> 00:32:15 I don't know if it'd be easy to compute it or not, maybe 513 00:32:15 --> 00:32:18 I won't tackle that here. 514 00:32:18 --> 00:32:22 But we could find the real part of that and the imaginary part 515 00:32:22 --> 00:32:30 of that, and we would have a genuine flow field. 516 00:32:30 --> 00:32:34 Satisfying Laplace's equation with the two orthogonal, the 517 00:32:34 --> 00:32:38 streamlines orthogonal to the equipotentials. 518 00:32:38 --> 00:32:41 We could totally do that example. 519 00:32:41 --> 00:32:43 OK, let me, yeah, thanks. 520 00:32:43 --> 00:32:48 AUDIENCE: You say one test question is based on x+iy? 521 00:32:49 --> 00:32:49 PROFESSOR STRANG: Yeah. 522 00:32:49 --> 00:32:50 Well, this sort of stuff. 523 00:32:50 --> 00:32:55 But, so u would be the, yeah that's right. 524 00:32:55 --> 00:33:02 Yeah, so a test question would be something like one way or 525 00:33:02 --> 00:33:08 another you would end up with a u, and an S, and the u+iS, if 526 00:33:08 --> 00:33:14 they're a good pair, would be some function of this magic z. 527 00:33:14 --> 00:33:16 Yeah, yeah. 528 00:33:16 --> 00:33:17 That's right. 529 00:33:17 --> 00:33:23 So whatever. 530 00:33:23 --> 00:33:25 We know examples, of course. 531 00:33:25 --> 00:33:30 For example, this could be x squared minus y squared. 532 00:33:30 --> 00:33:34 And the S that goes with that is 2xy, and the function that's 533 00:33:34 --> 00:33:43 involved there when I throw in the i is simply z squared. 534 00:33:43 --> 00:33:47 OK, that would be an example where the real and imaginary 535 00:33:47 --> 00:33:55 parts of this give us the good u, its good friend S, and the 536 00:33:55 --> 00:34:00 picture of stream lines and equipotentials meeting 537 00:34:00 --> 00:34:01 at right angles. 538 00:34:01 --> 00:34:06 Just, a beautiful picture, all coming out of this function. 539 00:34:06 --> 00:34:11 So probably the quiz will have some other function. 540 00:34:11 --> 00:34:20 But you'll still have a u and an S and a function of x+iy. 541 00:34:20 --> 00:34:23 So if it's not this one, which I don't think it is, it 542 00:34:23 --> 00:34:26 won't be be this one. 543 00:34:26 --> 00:34:29 AUDIENCE: [INAUDIBLE] 544 00:34:29 --> 00:34:29 PROFESSOR STRANG: Yes. 545 00:34:29 --> 00:34:34 Because first of all, I wouldn't have mentioned 546 00:34:34 --> 00:34:35 it if I was. 547 00:34:35 --> 00:34:40 And secondly, that's a little too messy, I think, to get a 548 00:34:40 --> 00:34:44 good handle of, to take the real and imaginary 549 00:34:44 --> 00:34:45 parts of that. 550 00:34:45 --> 00:34:47 It's not impossible, of course. 551 00:34:47 --> 00:34:49 We could completely do it. 552 00:34:49 --> 00:34:57 There'd be some ratio of two polynomials. 553 00:34:57 --> 00:35:02 Here we have just simple polynomials. 554 00:35:02 --> 00:35:04 OK, does that help with that question? 555 00:35:04 --> 00:35:05 Yeah. 556 00:35:05 --> 00:35:11 What else is on your mind? 557 00:35:11 --> 00:35:12 Any thoughts? 558 00:35:12 --> 00:35:14 Yeah, thanks. 559 00:35:14 --> 00:35:15 Curl, OK. 560 00:35:15 --> 00:35:21 Well, so I didn't really do three dimensions. 561 00:35:21 --> 00:35:26 But curl is important. 562 00:35:26 --> 00:35:33 And we did see in two dimensions the key fact that 563 00:35:33 --> 00:35:37 all this stuff, and let me just write down what the great 564 00:35:37 --> 00:35:39 connections are between these. 565 00:35:39 --> 00:35:42 Because I can't let the whole semester go without writing 566 00:35:42 --> 00:35:46 down that, what is it, the group? 567 00:35:46 --> 00:35:51 Is it the curl of a gradient? 568 00:35:51 --> 00:36:00 The curl of any gradient of u is always zero. 569 00:36:00 --> 00:36:01 Whatever it is. 570 00:36:01 --> 00:36:03 Whatever u is. 571 00:36:03 --> 00:36:09 And this comes from, let me put the other one down and 572 00:36:09 --> 00:36:14 then I'll just say why. 573 00:36:14 --> 00:36:17 The other one is like the transpose of this one. 574 00:36:17 --> 00:36:20 So if I transpose, so this is the zero operator. 575 00:36:20 --> 00:36:22 Curl times gradient gives the zero. 576 00:36:22 --> 00:36:25 So if I just transpose I still have zero. 577 00:36:25 --> 00:36:27 So if it's a transpose gradient, I have minus 578 00:36:27 --> 00:36:31 divergence, and actually if I transpose curl 579 00:36:31 --> 00:36:33 I get curl again. 580 00:36:33 --> 00:36:39 Of any, now I should put in, what should the curl act on? 581 00:36:39 --> 00:36:42 It acts on a w, I guess is. 582 00:36:42 --> 00:36:49 No, divergence w, it acts on an S, sorry. 583 00:36:49 --> 00:36:55 OK, and the minus sign, of course, isn't going to matter 584 00:36:55 --> 00:36:57 because I've got a zero on the right hand side. 585 00:36:57 --> 00:36:59 So S. 586 00:36:59 --> 00:37:04 Yeah, so if I take any field, I mean this is like, real 587 00:37:04 --> 00:37:08 proper vector calculus. 588 00:37:08 --> 00:37:13 To check these, I call them identities and maybe sometimes 589 00:37:13 --> 00:37:17 people indicate an identity meaning it's always true for 590 00:37:17 --> 00:37:19 every u, or for every S. 591 00:37:19 --> 00:37:21 They'll use the triple equals sign. 592 00:37:21 --> 00:37:24 Just to say they're really equal. 593 00:37:24 --> 00:37:34 OK, so we could define the curl, but you've met it 594 00:37:34 --> 00:37:39 elsewhere and maybe this isn't the time to do that. 595 00:37:39 --> 00:37:43 What's the key fact, the key little math business that 596 00:37:43 --> 00:37:45 makes all these true? 597 00:37:45 --> 00:37:52 So there's sort of a formal math fact that makes them true. 598 00:37:52 --> 00:37:58 And then there's the physical understanding of gradients 599 00:37:58 --> 00:38:03 being directions out with no rotation. 600 00:38:03 --> 00:38:06 So the physical understanding of that, but the math, the 601 00:38:06 --> 00:38:10 formal math fact is the fact that the second derivative 602 00:38:10 --> 00:38:16 of u over this vector x and y is equal to what? 603 00:38:16 --> 00:38:21 It's equal to second derivative of this vector y and x, yep. 604 00:38:21 --> 00:38:25 So you would find if you wrote out all the terms here, or all 605 00:38:25 --> 00:38:30 the terms here, you would find that just by using that fact, 606 00:38:30 --> 00:38:32 they all cancel each other. 607 00:38:32 --> 00:38:34 And the book, of course, does that. 608 00:38:34 --> 00:38:42 So we simply didn't do 3-D in the vector calculus section. 609 00:38:42 --> 00:38:45 So I'll stop there with that, because it's really 610 00:38:45 --> 00:38:50 saying that the curl is tremendously important. 611 00:38:50 --> 00:38:59 It measures vorticity and flow, and it's being able to. 612 00:38:59 --> 00:39:02 You know that like, you take the Navier-Stokes equations? 613 00:39:02 --> 00:39:10 Well, the pressure and the velocity are the, I'd say 614 00:39:10 --> 00:39:13 primary variables or the natural quantities to measure 615 00:39:13 --> 00:39:17 pressure and velocity for a fluid flow. 616 00:39:17 --> 00:39:23 But you could also use these identities to set up, in 617 00:39:23 --> 00:39:25 terms of other variables. 618 00:39:25 --> 00:39:32 Just rewrite the equation and you get other things. 619 00:39:32 --> 00:39:39 Mentioning Navier-Stokes and fluid flow reminds me to say, 620 00:39:39 --> 00:39:44 we keep using the example of Laplace's equation. 621 00:39:44 --> 00:39:48 And a person could say wait a minute, get beyond that. 622 00:39:48 --> 00:39:49 Right? 623 00:39:49 --> 00:39:54 So why are you always, when you teach finite elements, why do 624 00:39:54 --> 00:39:56 you always start with Laplace's equation? 625 00:39:56 --> 00:40:00 OK, well the main reason is it's the simplest one. 626 00:40:00 --> 00:40:03 It's the one where you can really see what's happening. 627 00:40:03 --> 00:40:07 More complicated equations would be for, like, elasticity, 628 00:40:07 --> 00:40:14 plane elasticity, or 3-D elasticity or other boundary 629 00:40:14 --> 00:40:17 value problems could be quite messy. 630 00:40:17 --> 00:40:23 But Laplace's equation is not totally a waste of time. 631 00:40:23 --> 00:40:26 First, it comes up when you have these scalar unknowns. 632 00:40:26 --> 00:40:31 And then it also comes up in numerical methods 633 00:40:31 --> 00:40:33 for Navier-Stokes. 634 00:40:33 --> 00:40:37 So the standard numerical method for Navier-Stokes, which 635 00:40:37 --> 00:40:43 would come in 18.086, ends up with Laplace's equation 636 00:40:43 --> 00:40:45 for the pressure. 637 00:40:45 --> 00:40:50 So to have a fast Laplace solver, as in today's 638 00:40:50 --> 00:40:53 lecture, pays off. 639 00:40:53 --> 00:40:57 So I'm just saying Laplace's equation is important in 640 00:40:57 --> 00:41:00 itself, it has the great advantage of being the 641 00:41:00 --> 00:41:02 simplest example we could possibly think of. 642 00:41:02 --> 00:41:06 It's the example where an x+iy trick works. 643 00:41:06 --> 00:41:11 And it actually comes up in serious big computations, 644 00:41:11 --> 00:41:16 because the equation for the pressure comes out to be a 645 00:41:16 --> 00:41:20 Laplace or a Poisson equation. 646 00:41:20 --> 00:41:21 Now. 647 00:41:21 --> 00:41:23 I kept going there. 648 00:41:23 --> 00:41:23 Yeah, thank you. 649 00:41:23 --> 00:41:28 AUDIENCE: [INAUDIBLE] 650 00:41:28 --> 00:41:31 PROFESSOR STRANG: We did, as a MATLAB problem. 651 00:41:31 --> 00:41:32 AUDIENCE: [INAUDIBLE] 652 00:41:32 --> 00:41:34 PROFESSOR STRANG: Sorry? 653 00:41:34 --> 00:41:35 And a first order, right. 654 00:41:35 --> 00:41:38 AUDIENCE: [INAUDIBLE] 655 00:41:38 --> 00:41:39 PROFESSOR STRANG: Huh. 656 00:41:39 --> 00:41:41 Yeah. 657 00:41:41 --> 00:41:43 I would do it the same way but it wouldn't be 658 00:41:43 --> 00:41:44 symmetric, of course. 659 00:41:44 --> 00:41:48 That was the point about that conviction term. 660 00:41:48 --> 00:41:50 Is, the diffusion term would be just what we've done, right? 661 00:41:50 --> 00:41:53 The diffusion with that second derivative. 662 00:41:53 --> 00:41:58 And what would it look like in, as long as we're close to, what 663 00:41:58 --> 00:42:05 would convection diffusion in 2-D look like? 664 00:42:05 --> 00:42:10 Just, I mean part of your interest is pass 18.085 665 00:42:10 --> 00:42:12 and get rid of it, right? 666 00:42:12 --> 00:42:20 But another part is like, these are problems that if you're in 667 00:42:20 --> 00:42:23 Course 16, Course 2, others, you're going to meet this. 668 00:42:23 --> 00:42:29 So the diffusion part is going to be, again in 2-D I'll have 669 00:42:29 --> 00:42:35 some minus u, well, I made it simple because I took 670 00:42:35 --> 00:42:36 c(x) to be one. 671 00:42:36 --> 00:42:39 It could have a c(x) in there. 672 00:42:39 --> 00:42:42 And what would the convection term look like? 673 00:42:42 --> 00:42:47 I'd have a velocity, in the x direction, of say a V_x. 674 00:42:49 --> 00:42:51 And a velocity in the y direction, V_y. 675 00:42:52 --> 00:42:56 V_y times, and that's just a number. 676 00:42:56 --> 00:43:01 In the simplest case that would be my river is traveling, or my 677 00:43:01 --> 00:43:10 flow is traveling along, and equals zero. 678 00:43:10 --> 00:43:10 AUDIENCE: [INAUDIBLE] 679 00:43:10 --> 00:43:12 PROFESSOR STRANG: Sorry? 680 00:43:12 --> 00:43:16 Yeah I don't know which way the river's traveling, actually. 681 00:43:16 --> 00:43:19 So those are just constants. 682 00:43:19 --> 00:43:23 They could have positive or negative signs. 683 00:43:23 --> 00:43:27 The V_x and V_y is the constant flow that's carrying, 684 00:43:27 --> 00:43:29 what am I doing here? 685 00:43:29 --> 00:43:33 The flow is flowing along, and if those are constants it's 686 00:43:33 --> 00:43:36 just flowing stead, steady flow. 687 00:43:36 --> 00:43:39 But it's diffusing at the same time. 688 00:43:39 --> 00:43:43 And this would bring in that same difficulties that we 689 00:43:43 --> 00:43:45 met in the MATLAB 1-D. 690 00:43:45 --> 00:43:50 So the MATLAB 1-D problem just didn't have a y. 691 00:43:50 --> 00:43:53 I don't care, yeah the sign I'm not worried about, it's just is 692 00:43:53 --> 00:43:56 that there, now I'm in 2-D. 693 00:43:56 --> 00:43:59 And what would I expect to see, I'd expect to see some 694 00:43:59 --> 00:44:01 trouble when V is large. 695 00:44:01 --> 00:44:06 When V is large, convection, this is convection down here. 696 00:44:06 --> 00:44:09 This is the convection part. 697 00:44:09 --> 00:44:15 And if V is large, then so that this should be a lower order 698 00:44:15 --> 00:44:19 term, is really fighting against this higher order term. 699 00:44:19 --> 00:44:22 I expect numerical difficulties, just 700 00:44:22 --> 00:44:24 the way we met. 701 00:44:24 --> 00:44:30 So anyway, if I did a MATLAB example, stretched it to 2-D 702 00:44:30 --> 00:44:34 we see a whole lot of interesting stuff. 703 00:44:34 --> 00:44:37 We'd see flow in, flow out, yeah. 704 00:44:37 --> 00:44:40 But I just can't do everything. 705 00:44:40 --> 00:44:43 But that would have a weak form, but your question about 706 00:44:43 --> 00:44:48 weak forms, weak form, when you have this anti-symmetric term 707 00:44:48 --> 00:44:52 for odd number of derivatives, is not quite as beautiful. 708 00:44:52 --> 00:44:55 But you have to deal with it, of course. 709 00:44:55 --> 00:44:56 Yep. 710 00:44:56 --> 00:45:01 OK, Ready for whatever. 711 00:45:01 --> 00:45:02 Any thoughts? 712 00:45:02 --> 00:45:08 Let's see, just have a look again at the list 713 00:45:08 --> 00:45:11 of problem topics. 714 00:45:11 --> 00:45:14 To see if anything occurs to you. 715 00:45:14 --> 00:45:16 I mean, not that it should. 716 00:45:16 --> 00:45:23 You know we're OK. 717 00:45:23 --> 00:45:29 I'm happy to call it a day on that, and time for dinner for 718 00:45:29 --> 00:45:31 everybody and see you at 7:30.