1 00:00:00,000 --> 00:00:00,499 2 00:00:00,499 --> 00:00:02,756 The following content is provided under a Creative 3 00:00:02,756 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,450 Your support will help MIT OpenCourseWare 5 00:00:05,450 --> 00:00:09,115 continue to offer high-quality educational resources for free. 6 00:00:09,115 --> 00:00:11,240 To make a donation, or to view additional materials 7 00:00:11,240 --> 00:00:15,870 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,870 --> 00:00:21,990 at ocw.mit.edu. 9 00:00:21,990 --> 00:00:24,750 PROFESSOR STRANG: So, shall we start, as always, 10 00:00:24,750 --> 00:00:26,570 just open for questions. 11 00:00:26,570 --> 00:00:29,150 About any topic. 12 00:00:29,150 --> 00:00:34,080 I listed again trusses, 1-D finite elements, 13 00:00:34,080 --> 00:00:35,540 grad, div, curl. 14 00:00:35,540 --> 00:00:39,970 And I should have squeezed in x+iy too, 15 00:00:39,970 --> 00:00:46,000 as the magic trick for finding solutions to Laplace's equation 16 00:00:46,000 --> 00:00:50,400 in 2-D. So those are all certainly topics that are 17 00:00:50,400 --> 00:00:52,210 in this part of the course. 18 00:00:52,210 --> 00:00:57,330 We didn't really get to 3-D, I'm sorry about that, 19 00:00:57,330 --> 00:00:59,160 where the curl comes in. 20 00:00:59,160 --> 00:01:01,900 Maybe I can say a few words about curl today. 21 00:01:01,900 --> 00:01:03,050 Anyway, questions. 22 00:01:03,050 --> 00:01:03,830 Discussion. 23 00:01:03,830 --> 00:01:04,530 Yes, thanks. 24 00:01:04,530 --> 00:01:10,600 AUDIENCE: [INAUDIBLE] 25 00:01:10,600 --> 00:01:12,520 PROFESSOR STRANG: Ooh, let's see. 26 00:01:12,520 --> 00:01:14,070 So A^T A for a truss. 27 00:01:14,070 --> 00:01:16,620 That's a good question. 28 00:01:16,620 --> 00:01:27,490 Trusses, A^T A. I guess I don't know any magic tricks either, 29 00:01:27,490 --> 00:01:32,420 so one way is to construct A, or A transpose, 30 00:01:32,420 --> 00:01:35,420 and then just multiply. 31 00:01:35,420 --> 00:01:43,290 A second way to do it would be by the four by four bar element 32 00:01:43,290 --> 00:01:47,700 matrices, so go bar by bar. 33 00:01:47,700 --> 00:01:54,650 So four by four bar matrices, four by four. 34 00:01:54,650 --> 00:02:00,270 So those are already in the A transpose A form. 35 00:02:00,270 --> 00:02:08,480 They're little A element, A bar transpose A, jeez, 36 00:02:08,480 --> 00:02:13,380 this isn't a great as it should be. 37 00:02:13,380 --> 00:02:20,010 A element, but I don't know if that would be, 38 00:02:20,010 --> 00:02:24,150 so and then you pop those into their correct places. 39 00:02:24,150 --> 00:02:28,880 I don't think I know any great idea beyond that. 40 00:02:28,880 --> 00:02:34,450 I think you should really be ready to construct a matrix A, 41 00:02:34,450 --> 00:02:35,890 yeah. 42 00:02:35,890 --> 00:02:44,250 For a reasonably small truss, of course. 43 00:02:44,250 --> 00:02:47,560 And of course the other part of trusses, 44 00:02:47,560 --> 00:02:56,070 the fun part is to be able to recognize solutions to Au=0. 45 00:02:56,070 --> 00:03:03,190 Possibly by looking at the truss more than by solving Au=0. 46 00:03:03,190 --> 00:03:05,700 Yeah, any particular example? 47 00:03:05,700 --> 00:03:10,790 Of a truss that I should look at just to pin this down? 48 00:03:10,790 --> 00:03:14,530 Any favorite trusses? 49 00:03:14,530 --> 00:03:19,150 There was an exam question, what was it, a complicated truss? 50 00:03:19,150 --> 00:03:22,210 Let's just create a truss. 51 00:03:22,210 --> 00:03:25,440 And just think about it. 52 00:03:25,440 --> 00:03:35,160 Maybe I won't create the whole matrix A. Here's a truss. 53 00:03:35,160 --> 00:03:37,650 How's that for a truss? 54 00:03:37,650 --> 00:03:43,430 So it's got-- And let me put no supports on it. 55 00:03:43,430 --> 00:03:48,260 Just, there's a truss to think about. 56 00:03:48,260 --> 00:03:52,290 Probably we won't get to all the gory details 57 00:03:52,290 --> 00:03:57,280 but if you look at that truss, what's the shape of A? 58 00:03:57,280 --> 00:04:06,680 Shape of the matrix A. I have a row for every bar. 59 00:04:06,680 --> 00:04:11,340 So one, two, three, four, five. 60 00:04:11,340 --> 00:04:15,470 And how many columns have I got, how many 61 00:04:15,470 --> 00:04:18,620 unknown u's, unknown displacements have I got? 62 00:04:18,620 --> 00:04:19,740 Eight. 63 00:04:19,740 --> 00:04:21,950 Two four, six, eight. 64 00:04:21,950 --> 00:04:30,980 So I would expect Au=0 would probably have how many 65 00:04:30,980 --> 00:04:32,960 independent solutions? 66 00:04:32,960 --> 00:04:36,540 Three. 67 00:04:36,540 --> 00:04:41,690 You don't know the exact rank, that's exactly true. 68 00:04:41,690 --> 00:04:43,880 There could be more than three, right. 69 00:04:43,880 --> 00:04:46,520 So to really pin it down you'd have 70 00:04:46,520 --> 00:04:51,370 to be sure you were right about that. 71 00:04:51,370 --> 00:04:54,050 So three, at least. 72 00:04:54,050 --> 00:05:01,800 And I guess here you could tell me the three solutions to Au=0. 73 00:05:01,800 --> 00:05:03,640 Three rigid motions. 74 00:05:03,640 --> 00:05:07,830 I could translate it to the right, I could translate it up, 75 00:05:07,830 --> 00:05:09,790 and you would know what the u is, 76 00:05:09,790 --> 00:05:16,910 so u translating to the right would be [1, 0, 1, 0, 1, 0, 1, 77 00:05:16,910 --> 00:05:19,360 0], right? 78 00:05:19,360 --> 00:05:25,370 That's horizontal motion all the same, rigid motion. 79 00:05:25,370 --> 00:05:29,200 And we should certainly discover that if we created A 80 00:05:29,200 --> 00:05:31,830 for this truss, that Au was zero. 81 00:05:31,830 --> 00:05:36,010 And similarly vertical motion and the third one would be? 82 00:05:36,010 --> 00:05:37,640 Rotation, rotation. 83 00:05:37,640 --> 00:05:39,690 Yeah. 84 00:05:39,690 --> 00:05:43,740 So if I did the rotation around there, for example, 85 00:05:43,740 --> 00:05:49,250 this guy would-- this u would also be a [1, 0] here. 86 00:05:49,250 --> 00:05:50,900 This wouldn't move. 87 00:05:50,900 --> 00:05:55,950 So I'm putting in the four pieces that would go into u. 88 00:05:55,950 --> 00:06:00,550 This one, what would be the u for this, the displacement 89 00:06:00,550 --> 00:06:05,600 of that corner of the truss? 90 00:06:05,600 --> 00:06:06,530 In a rotation? 91 00:06:06,530 --> 00:06:12,470 So my rotation is just swing this whole thing around. 92 00:06:12,470 --> 00:06:15,660 Zero? [0, -1], I think. 93 00:06:15,660 --> 00:06:16,470 Right. 94 00:06:16,470 --> 00:06:18,380 Because it's not going to go out, 95 00:06:18,380 --> 00:06:21,430 it's going to go straight down, [0, -1]. 96 00:06:21,430 --> 00:06:26,520 And what do you think this guy is? [1, -1], let's see. 97 00:06:26,520 --> 00:06:29,100 It's going to go this way, so it's 98 00:06:29,100 --> 00:06:31,020 going to go forward and down. 99 00:06:31,020 --> 00:06:32,570 And I think you're right. 100 00:06:32,570 --> 00:06:35,370 One and negative one, yeah. 101 00:06:35,370 --> 00:06:37,190 I think that would be right. 102 00:06:37,190 --> 00:06:46,310 Yeah, then the truss, we could check each bar. 103 00:06:46,310 --> 00:06:49,010 That bar, for example, should not change length 104 00:06:49,010 --> 00:06:53,630 because the movement is perpendicular to the bar 105 00:06:53,630 --> 00:06:55,830 and I'm writing ones, but I really 106 00:06:55,830 --> 00:07:00,550 should write some much smaller number like .1 everywhere, 107 00:07:00,550 --> 00:07:05,350 or something just so that this isn't a very big angle. 108 00:07:05,350 --> 00:07:09,910 And it wouldn't change length to first order. 109 00:07:09,910 --> 00:07:16,980 So that that's maybe an example where we see the motions, 110 00:07:16,980 --> 00:07:19,380 but we didn't actually create A, and we should 111 00:07:19,380 --> 00:07:23,420 be able to create A, don't let me prevent you 112 00:07:23,420 --> 00:07:28,380 from thinking about A. Yeah. 113 00:07:28,380 --> 00:07:30,929 AUDIENCE: [INAUDIBLE] Should those be ones, 114 00:07:30,929 --> 00:07:32,470 or should they be root two over twos? 115 00:07:32,470 --> 00:07:34,386 PROFESSOR STRANG: Well, that's a good question 116 00:07:34,386 --> 00:07:39,590 and after many years I've figured out that they're ones. 117 00:07:39,590 --> 00:07:41,730 But it's a very good question. 118 00:07:41,730 --> 00:07:45,000 Let's just see why. 119 00:07:45,000 --> 00:07:49,970 Let's look at this bar to be sure it's not stretched. 120 00:07:49,970 --> 00:07:50,570 Right? 121 00:07:50,570 --> 00:07:54,910 So this guy is moving over by one, and this also by one, 122 00:07:54,910 --> 00:07:55,980 is my claim. 123 00:07:55,980 --> 00:07:59,180 And then this movement down doesn't stretch it 124 00:07:59,180 --> 00:08:00,890 to first order, yeah. 125 00:08:00,890 --> 00:08:04,450 So I needed to make those guys the same. 126 00:08:04,450 --> 00:08:07,320 I guess what I figured out is that if you're rotating around 127 00:08:07,320 --> 00:08:14,340 here, then somehow it's the x and y, is that-- Anyway. 128 00:08:14,340 --> 00:08:19,330 Whatever. 129 00:08:19,330 --> 00:08:26,650 So that gives us a chance to do a specific example. 130 00:08:26,650 --> 00:08:31,070 OK, but I've dodged the creation of A. Yep, thanks. 131 00:08:31,070 --> 00:08:37,600 AUDIENCE: [INAUDIBLE] 132 00:08:37,600 --> 00:08:41,040 PROFESSOR STRANG: Sorry, the solution to A transpose A? 133 00:08:41,040 --> 00:08:46,424 AUDIENCE: [INAUDIBLE] 134 00:08:46,424 --> 00:08:47,840 PROFESSOR STRANG: Yeah, I see, OK. 135 00:08:47,840 --> 00:08:52,180 So the reason I stopped here was that A transpose A 136 00:08:52,180 --> 00:08:53,790 will be singular. 137 00:08:53,790 --> 00:09:00,440 So I wouldn't, like, go ahead, go forward to A transpose Au=f. 138 00:09:00,440 --> 00:09:08,430 But if I put on some supports, then of course now all good. 139 00:09:08,430 --> 00:09:14,840 So now I have, what's now the shape of A? 140 00:09:14,840 --> 00:09:17,210 For this one. 141 00:09:17,210 --> 00:09:22,270 I now have this bar is now, forget it, right? 142 00:09:22,270 --> 00:09:24,840 This bar is just between two supports. 143 00:09:24,840 --> 00:09:31,100 So if we put it in the matrix it'll just be a row of zeroes. 144 00:09:31,100 --> 00:09:33,520 Nothing will happen, and we're better off 145 00:09:33,520 --> 00:09:34,990 to just knock it out. 146 00:09:34,990 --> 00:09:39,470 So I think, now I have, I now have four bars. 147 00:09:39,470 --> 00:09:41,720 And how many unknowns? 148 00:09:41,720 --> 00:09:44,420 Four: two there and two there. 149 00:09:44,420 --> 00:09:48,340 And, do you guess that it's stable? 150 00:09:48,340 --> 00:09:49,570 That truss? 151 00:09:49,570 --> 00:09:51,780 Yeah, that looks stable to me. 152 00:09:51,780 --> 00:09:55,550 So the four by four matrix would be invertible 153 00:09:55,550 --> 00:09:58,050 and then I could solve. 154 00:09:58,050 --> 00:09:59,750 Good point. 155 00:09:59,750 --> 00:10:03,800 Then, A would be four by four, A transpose 156 00:10:03,800 --> 00:10:07,560 would be four by four, C would be four by four in between. 157 00:10:07,560 --> 00:10:11,450 This is the case that-- I gave a name for this case. 158 00:10:11,450 --> 00:10:14,770 When I have a square matrix, do you remember the name just 159 00:10:14,770 --> 00:10:15,900 for the hell of it? 160 00:10:15,900 --> 00:10:17,690 Statically determinate. 161 00:10:17,690 --> 00:10:21,520 It's determinate because each step determines everything 162 00:10:21,520 --> 00:10:22,580 completely. 163 00:10:22,580 --> 00:10:26,360 Normally, if I have another bar there, 164 00:10:26,360 --> 00:10:31,090 now it would be five by four, and now 165 00:10:31,090 --> 00:10:36,020 I really have to do the A transpose C A to get to a four 166 00:10:36,020 --> 00:10:38,590 by four invertible. 167 00:10:38,590 --> 00:10:43,350 By itself, A would not be invertible. 168 00:10:43,350 --> 00:10:47,230 This is the more typical case, where you really 169 00:10:47,230 --> 00:10:49,190 have to put all three together. 170 00:10:49,190 --> 00:10:51,410 Right. 171 00:10:51,410 --> 00:10:55,920 I hope you enjoyed the trusses part, though, and continue 172 00:10:55,920 --> 00:11:01,050 to enjoy them this evening. 173 00:11:01,050 --> 00:11:04,410 OK, I'll just keep moving to be sure 174 00:11:04,410 --> 00:11:07,220 that we cover any other topics. 175 00:11:07,220 --> 00:11:07,840 Yeah, thanks. 176 00:11:07,840 --> 00:11:09,834 AUDIENCE: [INAUDIBLE] 177 00:11:09,834 --> 00:11:11,250 PROFESSOR STRANG: For this matrix? 178 00:11:11,250 --> 00:11:13,036 AUDIENCE: [INAUDIBLE] 179 00:11:13,036 --> 00:11:14,410 PROFESSOR STRANG: For that truss? 180 00:11:14,410 --> 00:11:15,960 Oh my God. 181 00:11:15,960 --> 00:11:20,260 OK, let me see. 182 00:11:20,260 --> 00:11:23,680 Then can I do one row? 183 00:11:23,680 --> 00:11:27,990 OK, of course, you guys are responsible for much more. 184 00:11:27,990 --> 00:11:29,310 Alright, which row shall I do? 185 00:11:29,310 --> 00:11:32,310 Which bar? 186 00:11:32,310 --> 00:11:33,430 A diagonal bar? 187 00:11:33,430 --> 00:11:38,210 I knew you'd make it like, you could make up quizzes 188 00:11:38,210 --> 00:11:40,180 and I could just sit back here. 189 00:11:40,180 --> 00:11:43,860 OK, so let's take this diagonal bar, alright. 190 00:11:43,860 --> 00:11:47,470 And are we going to keep that supported, or not? 191 00:11:47,470 --> 00:11:49,590 Do you want to keep it supported? 192 00:11:49,590 --> 00:11:53,530 OK, so in this case then that end is not moving. 193 00:11:53,530 --> 00:11:55,960 So this will be in this, and this bar 194 00:11:55,960 --> 00:12:01,470 corresponds to a row of A. And how many non-zeroes will 195 00:12:01,470 --> 00:12:03,540 I expect in that row? 196 00:12:03,540 --> 00:12:04,750 Just two. 197 00:12:04,750 --> 00:12:08,410 Normally four, but I'm not getting any motion down here. 198 00:12:08,410 --> 00:12:11,500 So it'll just be two and if that's 45 degrees, 199 00:12:11,500 --> 00:12:16,370 shall we say, then that row, I think, would be what? 200 00:12:16,370 --> 00:12:19,320 Well, OK. 201 00:12:19,320 --> 00:12:21,680 Where do my non-zeroes appear? 202 00:12:21,680 --> 00:12:26,080 This is node number one with an H and a V. 203 00:12:26,080 --> 00:12:30,910 So I think we have zeroes. 204 00:12:30,910 --> 00:12:37,400 If that bar stretches-- The connection between displacement 205 00:12:37,400 --> 00:12:40,840 and stretching of this bar does not involve this guy. 206 00:12:40,840 --> 00:12:42,800 So I think it's zero and zero there. 207 00:12:42,800 --> 00:12:45,350 And now, what else is it? 208 00:12:45,350 --> 00:12:47,200 So now come the real numbers, which 209 00:12:47,200 --> 00:12:51,320 I believe to be cosine and sine of that angle. 210 00:12:51,320 --> 00:12:56,400 Because if I, how much does that bar stretch? 211 00:12:56,400 --> 00:12:59,865 I know that I'm looking for a cosine and a sine, 212 00:12:59,865 --> 00:13:04,840 and if this goes positively, then the bar does stretch. 213 00:13:04,840 --> 00:13:08,060 If this goes positively, that does stretch the bar 214 00:13:08,060 --> 00:13:10,810 so I'm expecting positive numbers there, 215 00:13:10,810 --> 00:13:14,270 like the cosine, square root of two over two, 216 00:13:14,270 --> 00:13:20,360 and the sine, square root of two over two. 217 00:13:20,360 --> 00:13:25,670 Well, I dodged the bullet of getting the whole matrix, 218 00:13:25,670 --> 00:13:27,680 but maybe that would do it. 219 00:13:27,680 --> 00:13:30,280 Why don't we do this the top one? 220 00:13:30,280 --> 00:13:31,790 Yeah. 221 00:13:31,790 --> 00:13:36,370 Tell me the first, if that's bar one, what would 222 00:13:36,370 --> 00:13:38,950 be the first row of the matrix? 223 00:13:38,950 --> 00:13:42,410 OK, it involves both of these nodes. 224 00:13:42,410 --> 00:13:44,080 But the angle is zero. 225 00:13:44,080 --> 00:13:48,620 So that's going to be a little special. 226 00:13:48,620 --> 00:13:55,550 So if this goes out horizontally, 227 00:13:55,550 --> 00:13:57,350 I should really start with the first one. 228 00:13:57,350 --> 00:14:00,160 Is this goes horizontally it compresses the bar, 229 00:14:00,160 --> 00:14:02,500 I think we get a minus one there. 230 00:14:02,500 --> 00:14:05,670 If it goes vertically, that doesn't do anything. 231 00:14:05,670 --> 00:14:08,430 If this goes horizontally it does do something. 232 00:14:08,430 --> 00:14:11,090 If it goes vertically it doesn't. 233 00:14:11,090 --> 00:14:22,370 I'd say that would be the row of the matrix coming from the top. 234 00:14:22,370 --> 00:14:24,100 That would give me the stretching. 235 00:14:24,100 --> 00:14:25,840 You remember, I'm always going to, 236 00:14:25,840 --> 00:14:28,490 I think of multiplying this by u. 237 00:14:28,490 --> 00:14:37,200 I think of multiplying that by [u 1 H, u 1 V, u 2 H, u 2 V], 238 00:14:37,200 --> 00:14:41,960 and this top row should give me you 239 00:14:41,960 --> 00:14:57,930 u 2 H minus u 1 H, which is the stretch in bar one. 240 00:14:57,930 --> 00:15:00,470 So that would be a typical one, this 241 00:15:00,470 --> 00:15:03,870 would be at least typical of one where 242 00:15:03,870 --> 00:15:06,500 I do see a cosine and a sine. 243 00:15:06,500 --> 00:15:15,270 And let me just finally add, suppose this was not supported. 244 00:15:15,270 --> 00:15:17,280 OK, suppose that's not supported, 245 00:15:17,280 --> 00:15:20,430 now I've got a couple more columns to squeeze in. 246 00:15:20,430 --> 00:15:23,220 Maybe I can somehow do it here. 247 00:15:23,220 --> 00:15:25,820 Can I squeeze in the two more columns? 248 00:15:25,820 --> 00:15:31,240 So can you complete the top row of the matrix? 249 00:15:31,240 --> 00:15:33,310 Now I've got six columns. 250 00:15:33,310 --> 00:15:37,550 Because here's two, here's two, here's two more. 251 00:15:37,550 --> 00:15:40,850 What goes on the top row of a matrix now? 252 00:15:40,850 --> 00:15:44,580 Zeroes, because this is not affected by bar one. 253 00:15:44,580 --> 00:15:47,920 But it is affected by this bar. 254 00:15:47,920 --> 00:15:52,070 So it's going to show up in this row, and how will it show up? 255 00:15:52,070 --> 00:15:53,290 Two negatives, right. 256 00:15:53,290 --> 00:15:59,450 A negative cosine and a negative sine. and at 45 degrees I 257 00:15:59,450 --> 00:16:02,220 can't tell the difference. 258 00:16:02,220 --> 00:16:05,940 Because if these move forward, that compresses the bar. 259 00:16:05,940 --> 00:16:07,210 So the minus sign. 260 00:16:07,210 --> 00:16:11,430 So again, the rows add up to zero, 261 00:16:11,430 --> 00:16:18,220 as we expect when the bar is not touching a support. 262 00:16:18,220 --> 00:16:26,240 This is not touching a support, so it adds up to zero. 263 00:16:26,240 --> 00:16:34,840 OK, we'll have a truss problem this evening, 264 00:16:34,840 --> 00:16:39,500 but not a big messy one. 265 00:16:39,500 --> 00:16:42,320 How about finite elements? 266 00:16:42,320 --> 00:16:45,000 You guys, do you like finite elements? 267 00:16:45,000 --> 00:16:49,180 I'm sort of hoping to make them attractive. 268 00:16:49,180 --> 00:16:52,960 I noticed a problem, just to give us some specific one 269 00:16:52,960 --> 00:16:56,850 to work on, and I don't remember that it was a homework problem. 270 00:16:56,850 --> 00:17:07,590 This is Section 3.1, number 18, asks about the equation u''=0. 271 00:17:07,590 --> 00:17:11,260 Well, we've talked about it in class. 272 00:17:11,260 --> 00:17:19,220 With u(0)=0 but u' of-- the slope equal zero at the other 273 00:17:19,220 --> 00:17:19,860 end. 274 00:17:19,860 --> 00:17:23,130 So what's the picture if I use linear elements? 275 00:17:23,130 --> 00:17:26,300 I don't remember how many I used in the problem. 276 00:17:26,300 --> 00:17:34,500 Well, it allows you to use N interior guys, one, 277 00:17:34,500 --> 00:17:41,060 two, up to N, and then another-- This will come in. 278 00:17:41,060 --> 00:17:45,710 Or that's the point. 279 00:17:45,710 --> 00:17:49,340 OK, so what's the finite element method, 280 00:17:49,340 --> 00:17:52,030 finite element matrix K for this-- 281 00:17:52,030 --> 00:18:01,360 So I want to do linear elements and I want to construct K. 282 00:18:01,360 --> 00:18:03,880 And, yeah, I guess. 283 00:18:03,880 --> 00:18:07,580 Oh, I haven't actually made anything 284 00:18:07,580 --> 00:18:09,370 happen to this problem. 285 00:18:09,370 --> 00:18:12,970 All zeroes is kind of slow going. 286 00:18:12,970 --> 00:18:16,840 u will be, the solution will certainly be zero. 287 00:18:16,840 --> 00:18:24,800 So maybe I'd better put in a load here to get some action. 288 00:18:24,800 --> 00:18:32,350 OK, well, yeah. 289 00:18:32,350 --> 00:18:36,430 So I proposed this question but now, is this a question 290 00:18:36,430 --> 00:18:38,610 to think about? 291 00:18:38,610 --> 00:18:41,490 I think that's a reasonable example to do. 292 00:18:41,490 --> 00:18:43,990 It's got the two types of boundary conditions. 293 00:18:43,990 --> 00:18:46,100 It's got the right-hand side f, it's 294 00:18:46,100 --> 00:18:51,440 got linear elements which means it's kind of doable by hand. 295 00:18:51,440 --> 00:19:00,760 And we kind of know what matrix to expect out of it. 296 00:19:00,760 --> 00:19:05,280 What matrix do we expect? 297 00:19:05,280 --> 00:19:08,370 What do I expect out of linear elements, 298 00:19:08,370 --> 00:19:11,690 do you remember the point about linear elements on equally 299 00:19:11,690 --> 00:19:13,300 spaced meshes? 300 00:19:13,300 --> 00:19:17,510 That just brought back our regular difference matrices. 301 00:19:17,510 --> 00:19:21,300 So I'm expecting this stiffness matrix 302 00:19:21,300 --> 00:19:22,620 to be a difference matrix. 303 00:19:22,620 --> 00:19:24,890 Anyway, the point of this question 304 00:19:24,890 --> 00:19:28,290 is, OK, I have a hat function, I have a hat function, 305 00:19:28,290 --> 00:19:34,940 I've a hat function, a hat function, and is that the end? 306 00:19:34,940 --> 00:19:38,560 Is that the complete list of my trial functions? 307 00:19:38,560 --> 00:19:40,660 One more, right? 308 00:19:40,660 --> 00:19:48,530 Because this condition is, all my trial and test functions 309 00:19:48,530 --> 00:19:51,330 don't have to satisfy this. 310 00:19:51,330 --> 00:19:55,580 So I'm allowed, and should have, another guy there. 311 00:19:55,580 --> 00:19:58,530 A half hat for that one. 312 00:19:58,530 --> 00:20:06,350 You may say, don't let that clown into the finite element 313 00:20:06,350 --> 00:20:09,080 space but I think it should be. 314 00:20:09,080 --> 00:20:13,900 The solution won't use much of it. 315 00:20:13,900 --> 00:20:18,450 Because the solution is going to aim for zero slope. 316 00:20:18,450 --> 00:20:20,850 But it's going to need a little-- You see why it needs 317 00:20:20,850 --> 00:20:22,360 a little bit, something here? 318 00:20:22,360 --> 00:20:26,150 Because this thing has slope down. 319 00:20:26,150 --> 00:20:29,140 So if there's some of that in there, 320 00:20:29,140 --> 00:20:35,390 there better be somebody else to cancel it. 321 00:20:35,390 --> 00:20:39,580 If our approximation is going to have about zero slope. 322 00:20:39,580 --> 00:20:41,340 OK, so then. 323 00:20:41,340 --> 00:20:44,000 Can you construct a matrix K? 324 00:20:44,000 --> 00:20:48,110 Let's see, what's the (2,3) entry? 325 00:20:48,110 --> 00:20:52,487 So if I call this number one, this number two, 326 00:20:52,487 --> 00:20:53,570 oh, I've already numbered. 327 00:20:53,570 --> 00:20:57,730 So number two and number three, so that trial 328 00:20:57,730 --> 00:20:59,790 function against that one. 329 00:20:59,790 --> 00:21:00,450 What do I? 330 00:21:00,450 --> 00:21:05,340 What's my formula for the (2,3) entry of the stiffness matrix? 331 00:21:05,340 --> 00:21:08,690 It's some integral, right? 332 00:21:08,690 --> 00:21:10,920 And what do I integrate? 333 00:21:10,920 --> 00:21:12,830 I integrate, yeah. 334 00:21:12,830 --> 00:21:15,560 And I've got to have to remember. 335 00:21:15,560 --> 00:21:22,470 So I do, yeah, my weak form-- I've integrated by parts, 336 00:21:22,470 --> 00:21:28,670 so my weak form is the integral of u'*v'*dx equals the integral 337 00:21:28,670 --> 00:21:33,010 of f times. v dx. 338 00:21:33,010 --> 00:21:34,120 That's my weak form. 339 00:21:34,120 --> 00:21:41,070 I did two integrations by parts and the integrated term 340 00:21:41,070 --> 00:21:41,870 will go away. 341 00:21:41,870 --> 00:21:43,430 Because of those zeroes. 342 00:21:43,430 --> 00:21:52,020 OK, so K_(2,3) will come from this side when I'm using phi_2 343 00:21:52,020 --> 00:21:55,380 and phi_3, because I'm taking the phis, 344 00:21:55,380 --> 00:22:00,460 the phis and the V's both the same hat function. 345 00:22:00,460 --> 00:22:08,270 OK, so what do I get for that? phi_2' is? 346 00:22:08,270 --> 00:22:12,030 So this is it, and it overlaps this one. 347 00:22:12,030 --> 00:22:15,290 So when it overlaps this phi_2 is coming down 348 00:22:15,290 --> 00:22:17,480 and phi_3 is going up. 349 00:22:17,480 --> 00:22:23,210 And the slope is 1/h, let's say. 350 00:22:23,210 --> 00:22:30,240 So I think I'm integrating us a negative slope, is that right? 351 00:22:30,240 --> 00:22:32,840 Times a positive slope. 352 00:22:32,840 --> 00:22:38,960 And I'm really only integrating over one h interval. 353 00:22:38,960 --> 00:22:44,550 The two overlap only here, where this one's coming down 354 00:22:44,550 --> 00:22:46,020 and that one's going up. 355 00:22:46,020 --> 00:22:49,920 So I think-- dx. 356 00:22:49,920 --> 00:22:52,170 And the great thing is, of course, we have a constant. 357 00:22:52,170 --> 00:22:54,880 So I have minus one over h squared times h, 358 00:22:54,880 --> 00:22:58,230 I think minus one over h. 359 00:22:58,230 --> 00:23:00,990 That would be K_(2,3). 360 00:23:00,990 --> 00:23:08,070 That's a simple example. 361 00:23:08,070 --> 00:23:19,050 And then at the end we will see it, we'll see this one, 362 00:23:19,050 --> 00:23:21,850 I think we'll get some matrix. 363 00:23:21,850 --> 00:23:24,340 We'll have this 1/h outside, I think 364 00:23:24,340 --> 00:23:27,980 we'll have something like two, minus one; two, minus one; 365 00:23:27,980 --> 00:23:31,330 minus one and then only a one from the half-hat. 366 00:23:31,330 --> 00:23:39,190 I think it would be that matrix that would be K. I think. 367 00:23:39,190 --> 00:23:42,440 Maybe with more, greater size if we 368 00:23:42,440 --> 00:23:43,920 have a whole bunch of elements. 369 00:23:43,920 --> 00:23:47,590 But that pattern. 370 00:23:47,590 --> 00:23:49,260 You're pretty much into this? 371 00:23:49,260 --> 00:23:54,420 Yeah, I mean we're doing a lot in this course. 372 00:23:54,420 --> 00:23:59,090 I'm really grateful you guys stay with it, 373 00:23:59,090 --> 00:24:06,770 and kept to these new ideas, through doing exercises 374 00:24:06,770 --> 00:24:07,420 and so on. 375 00:24:07,420 --> 00:24:11,510 Because there's a lot here. 376 00:24:11,510 --> 00:24:13,440 Well, I thought I'd put an example up, 377 00:24:13,440 --> 00:24:16,650 to open up, just to remind you what 378 00:24:16,650 --> 00:24:18,360 that language is about there. 379 00:24:18,360 --> 00:24:23,340 And to be ready for any question in that topic. 380 00:24:23,340 --> 00:24:26,600 Or any question whatever. 381 00:24:26,600 --> 00:24:28,990 So I jumped in with finite elements, 382 00:24:28,990 --> 00:24:37,320 but I'm ready also to talk about that area of the course. 383 00:24:37,320 --> 00:24:38,470 AUDIENCE: [INAUDIBLE] 384 00:24:38,470 --> 00:24:44,630 PROFESSOR STRANG: Yeah. x+iy stuff, OK. 385 00:24:44,630 --> 00:24:47,940 Basically, any function of x+iy, yeah. 386 00:24:47,940 --> 00:24:49,570 Any function. 387 00:24:49,570 --> 00:24:53,630 So strictly, yeah, I mean a mathematician would say what, 388 00:24:53,630 --> 00:24:54,640 any function? 389 00:24:54,640 --> 00:25:02,200 That's, you've opened the door to crazy things saying that. 390 00:25:02,200 --> 00:25:07,700 So what I really mean is, we have these powers of x+iy, 391 00:25:07,700 --> 00:25:09,840 and then we have combinations of them. 392 00:25:09,840 --> 00:25:12,430 So the only requirement would be that if I 393 00:25:12,430 --> 00:25:14,760 want to take an infinite combination 394 00:25:14,760 --> 00:25:19,170 it should, the series should, add up to something. 395 00:25:19,170 --> 00:25:23,180 If it has a nice Taylor series then 396 00:25:23,180 --> 00:25:25,450 those are the best functions there are. 397 00:25:25,450 --> 00:25:27,730 Functions with nice Taylor series. 398 00:25:27,730 --> 00:25:29,290 I'll just say it. 399 00:25:29,290 --> 00:25:30,880 Having used those words. 400 00:25:30,880 --> 00:25:37,130 Suppose I take that function. 401 00:25:37,130 --> 00:25:39,850 There's a function, that's a function-- z is x+iy. 402 00:25:39,850 --> 00:25:46,270 403 00:25:46,270 --> 00:25:50,160 But z is shorter to write. 404 00:25:50,160 --> 00:25:52,640 So it's not a polynomial, obviously. 405 00:25:52,640 --> 00:25:54,770 But it is a function of x+iy. 406 00:25:54,770 --> 00:26:00,620 407 00:26:00,620 --> 00:26:02,180 Well, tell me this. 408 00:26:02,180 --> 00:26:05,510 Where does that function go wrong? 409 00:26:05,510 --> 00:26:10,820 So e^z is a function that never goes wrong, right? e^z, 410 00:26:10,820 --> 00:26:14,270 that series always converges. 411 00:26:14,270 --> 00:26:18,600 Can you tell me the series, if I expand that into a series, 412 00:26:18,600 --> 00:26:22,960 what series am I looking at? 413 00:26:22,960 --> 00:26:28,900 This is not on the exam, so to speak. 414 00:26:28,900 --> 00:26:32,130 Do you know one over one plus something, 415 00:26:32,130 --> 00:26:39,730 what's the series for that? 416 00:26:39,730 --> 00:26:43,370 Well the constant term, when z is zero is certainly a one. 417 00:26:43,370 --> 00:26:47,330 I think the trick, it's this is geometric series, 418 00:26:47,330 --> 00:26:51,500 and because it's a z squared it's that. 419 00:26:51,500 --> 00:26:56,160 That would be the geometric series. 420 00:26:56,160 --> 00:26:58,630 With constant ratio z squared. 421 00:26:58,630 --> 00:27:02,350 If I multiply that by that, 1 plus z 422 00:27:02,350 --> 00:27:04,330 squared times that, everything will cancel 423 00:27:04,330 --> 00:27:05,840 and I'll get the one. 424 00:27:05,840 --> 00:27:08,800 That's it. 425 00:27:08,800 --> 00:27:14,550 So there is a Taylor series for this function. 426 00:27:14,550 --> 00:27:19,780 Now, the reason I chose that example is, you could tell me, 427 00:27:19,780 --> 00:27:23,490 it doesn't converge if z is too large, right? 428 00:27:23,490 --> 00:27:28,040 Is this an analytic function? 429 00:27:28,040 --> 00:27:30,840 Where is this a good function and where does it 430 00:27:30,840 --> 00:27:33,030 have problems? 431 00:27:33,030 --> 00:27:38,540 If z is less than one, and I really mean magnitude of z, 432 00:27:38,540 --> 00:27:40,810 so let me draw the z-plane. 433 00:27:40,810 --> 00:27:44,910 Here's the real part of z that you usually call x, 434 00:27:44,910 --> 00:27:49,370 and the imaginary part of z that you usually call y, 435 00:27:49,370 --> 00:27:55,190 because z is x+iy, and where will this series converge? 436 00:27:55,190 --> 00:28:01,840 It'll converge out as far as this circle. 437 00:28:01,840 --> 00:28:05,210 This is the Taylor series around zero. 438 00:28:05,210 --> 00:28:05,710 Right? 439 00:28:05,710 --> 00:28:10,640 The constant term I found at z=0. 440 00:28:10,640 --> 00:28:15,550 Then that series, this function, is great. 441 00:28:15,550 --> 00:28:18,860 It's an analytic function, everything, 442 00:28:18,860 --> 00:28:22,480 it gives us a solution to Laplace's-- This'll be, 443 00:28:22,480 --> 00:28:26,530 the real and imaginary parts of that will be the u and the s 444 00:28:26,530 --> 00:28:28,930 that solve Laplace's equation. 445 00:28:28,930 --> 00:28:33,450 Out to, at least in this circle. 446 00:28:33,450 --> 00:28:38,180 But something, there's a problem at the edge of the circle. 447 00:28:38,180 --> 00:28:42,400 Now, here's my point. 448 00:28:42,400 --> 00:28:47,550 If I think of one over one plus x squared, 449 00:28:47,550 --> 00:28:49,310 look at that for a minute. 450 00:28:49,310 --> 00:28:51,930 That function has no problems at all, right? 451 00:28:51,930 --> 00:28:55,500 One over one plus x squared, you can let x be anything? 452 00:28:55,500 --> 00:28:58,880 It's no trouble. 453 00:28:58,880 --> 00:29:00,840 But one over one plus z squared, when 454 00:29:00,840 --> 00:29:04,190 we look in the complex plane, ah, we find a problem. 455 00:29:04,190 --> 00:29:07,740 And where is the problem with this function? 456 00:29:07,740 --> 00:29:12,530 At z equals, so everybody's looking at this guy. 457 00:29:12,530 --> 00:29:16,730 There's a problem with that function at z=i. 458 00:29:16,730 --> 00:29:20,200 And it happens to be not an accident, 459 00:29:20,200 --> 00:29:22,430 it's right there on the circle. 460 00:29:22,430 --> 00:29:26,820 It's hiding in the complex-- it's not on the real axis. 461 00:29:26,820 --> 00:29:29,630 So the real person didn't notice it. 462 00:29:29,630 --> 00:29:33,280 But the complex person said ah, that's the problem. 463 00:29:33,280 --> 00:29:37,310 There's a singularity there, and of course it's called a pole, 464 00:29:37,310 --> 00:29:42,840 and people in so many parts of science are interested in that. 465 00:29:42,840 --> 00:29:46,500 Is there any other place that there's a problem? 466 00:29:46,500 --> 00:29:47,990 At minus i. 467 00:29:47,990 --> 00:29:50,870 When z is minus i we'll also get a problem. 468 00:29:50,870 --> 00:29:56,760 So this is a function with two poles, those two poles 469 00:29:56,760 --> 00:30:05,510 and they're the reason that the series couldn't make it. 470 00:30:05,510 --> 00:30:09,330 Going out this way the series doesn't meet any problems. 471 00:30:09,330 --> 00:30:14,170 But the series always goes out in a circle, 472 00:30:14,170 --> 00:30:18,150 and the first circle, the first guy, the first problem it hits, 473 00:30:18,150 --> 00:30:21,480 the series stops converging. 474 00:30:21,480 --> 00:30:24,400 By the way, let me ask you a question. 475 00:30:24,400 --> 00:30:29,760 Suppose I instead did the Taylor series around this point? 476 00:30:29,760 --> 00:30:31,680 Now, what do I mean by that? 477 00:30:31,680 --> 00:30:33,630 That's the point one, let's say. 478 00:30:33,630 --> 00:30:36,620 What do I mean by that, the Taylor series around one? 479 00:30:36,620 --> 00:30:41,820 I'll rewrite the function as one plus, well now, what do I do? 480 00:30:41,820 --> 00:30:46,680 I want it in z minus one squared. 481 00:30:46,680 --> 00:30:47,720 Oh, gosh. 482 00:30:47,720 --> 00:30:54,660 I'm getting beyond what you will care about. 483 00:30:54,660 --> 00:31:00,820 Again, if I expanded, if I wrote the power series in powers of z 484 00:31:00,820 --> 00:31:05,490 minus one, what would it work in? 485 00:31:05,490 --> 00:31:07,900 And then I'll stop with this example. 486 00:31:07,900 --> 00:31:12,950 The circle would reach out until it hit a pole. 487 00:31:12,950 --> 00:31:14,700 And it can't make it past that pole. 488 00:31:14,700 --> 00:31:18,630 So it would be a circle of radius square root of two, 489 00:31:18,630 --> 00:31:20,700 there would be a circle there. 490 00:31:20,700 --> 00:31:24,690 If we were going to discuss, and this is really 491 00:31:24,690 --> 00:31:26,050 Chapter 5 of the book. 492 00:31:26,050 --> 00:31:34,030 I mention it, because you've got a book that explains this. 493 00:31:34,030 --> 00:31:36,690 If I thought the center of the universe was there, 494 00:31:36,690 --> 00:31:39,470 and then the poles are still here, 495 00:31:39,470 --> 00:31:42,370 the circle will make it out to those poles. 496 00:31:42,370 --> 00:31:46,870 So I can do Taylor series, I can sort of hook together Taylor 497 00:31:46,870 --> 00:31:49,200 series all over the place. 498 00:31:49,200 --> 00:31:52,470 And they'll all quit when they reach a pole, 499 00:31:52,470 --> 00:31:54,380 but when I put all those circles together 500 00:31:54,380 --> 00:31:57,990 I can get all the rest of the plane. 501 00:31:57,990 --> 00:32:01,750 OK, so that's something about, I don't 502 00:32:01,750 --> 00:32:06,220 know how I got onto that department, but it's amazing. 503 00:32:06,220 --> 00:32:10,580 This, so the real and imaginary parts of that would be a flow, 504 00:32:10,580 --> 00:32:12,080 would give me a flow. 505 00:32:12,080 --> 00:32:15,390 I don't know if it'd be easy to compute it or not, 506 00:32:15,390 --> 00:32:18,310 maybe I won't tackle that here. 507 00:32:18,310 --> 00:32:21,240 But we could find the real part of that 508 00:32:21,240 --> 00:32:24,280 and the imaginary part of that, and we would 509 00:32:24,280 --> 00:32:30,420 have a genuine flow field. 510 00:32:30,420 --> 00:32:34,060 Satisfying Laplace's equation with the two orthogonal, 511 00:32:34,060 --> 00:32:38,550 the streamlines orthogonal to the equipotentials. 512 00:32:38,550 --> 00:32:41,840 We could totally do that example. 513 00:32:41,840 --> 00:32:43,480 OK, let me, yeah, thanks. 514 00:32:43,480 --> 00:32:48,378 AUDIENCE: You say one test question is based on x+iy? 515 00:32:48,378 --> 00:32:49,336 PROFESSOR STRANG: Yeah. 516 00:32:49,336 --> 00:32:50,377 Well, this sort of stuff. 517 00:32:50,377 --> 00:32:55,180 But, so u would be the, yeah that's right. 518 00:32:55,180 --> 00:33:01,480 Yeah, so a test question would be something like, 519 00:33:01,480 --> 00:33:06,630 one way or another, you would end up with a u, and an s, 520 00:33:06,630 --> 00:33:10,200 and the u+is, if they're a good pair, 521 00:33:10,200 --> 00:33:14,370 would be some function of this magic z. 522 00:33:14,370 --> 00:33:16,609 Yeah, yeah. 523 00:33:16,609 --> 00:33:17,150 That's right. 524 00:33:17,150 --> 00:33:23,140 So whatever. 525 00:33:23,140 --> 00:33:25,670 We know examples, of course. 526 00:33:25,670 --> 00:33:30,050 For example, this could be x squared minus y squared. 527 00:33:30,050 --> 00:33:33,840 And the s that goes with that is 2xy, 528 00:33:33,840 --> 00:33:38,550 and the function that's involved there when I throw in the i 529 00:33:38,550 --> 00:33:43,020 is simply z squared. 530 00:33:43,020 --> 00:33:45,750 OK, that would be an example where 531 00:33:45,750 --> 00:33:50,640 the real and imaginary parts of this give us the good u, 532 00:33:50,640 --> 00:33:55,810 its good friend s, and the picture 533 00:33:55,810 --> 00:33:59,880 of streamlines and equipotentials 534 00:33:59,880 --> 00:34:01,820 meeting at right angles. 535 00:34:01,820 --> 00:34:06,880 Just, a beautiful picture, all coming out of this function. 536 00:34:06,880 --> 00:34:11,990 So probably the quiz will have some other function. 537 00:34:11,990 --> 00:34:20,250 But you'll still have a u and an s and a function of x+iy. 538 00:34:20,250 --> 00:34:23,750 So if it's not this one, which I don't think it is. 539 00:34:23,750 --> 00:34:26,480 It won't be be this one. 540 00:34:26,480 --> 00:34:29,034 AUDIENCE: [INAUDIBLE] 541 00:34:29,034 --> 00:34:29,950 PROFESSOR STRANG: Yes. 542 00:34:29,950 --> 00:34:35,800 Because first of all, I wouldn't have mentioned it if I was. 543 00:34:35,800 --> 00:34:39,010 And secondly, that's a little too messy, 544 00:34:39,010 --> 00:34:43,220 I think, to get a good handle of, 545 00:34:43,220 --> 00:34:45,360 to take the real and imaginary parts of that. 546 00:34:45,360 --> 00:34:47,400 It's not impossible, of course. 547 00:34:47,400 --> 00:34:49,210 We could completely do it. 548 00:34:49,210 --> 00:34:57,120 There'd be some ratio of two polynomials. 549 00:34:57,120 --> 00:35:02,380 Here we have just simple polynomials. 550 00:35:02,380 --> 00:35:04,620 OK, does that help with that question? 551 00:35:04,620 --> 00:35:05,500 Yeah. 552 00:35:05,500 --> 00:35:11,040 What else is on your mind? 553 00:35:11,040 --> 00:35:12,380 Any thoughts? 554 00:35:12,380 --> 00:35:14,100 Yeah, thanks. 555 00:35:14,100 --> 00:35:15,490 Curl, OK. 556 00:35:15,490 --> 00:35:21,520 Well, so I didn't really do three dimensions. 557 00:35:21,520 --> 00:35:26,940 But curl is important. 558 00:35:26,940 --> 00:35:31,770 And we did see, in two dimensions the key fact 559 00:35:31,770 --> 00:35:35,410 that all this stuff-- Let me just 560 00:35:35,410 --> 00:35:39,120 write down what the great connections are between these. 561 00:35:39,120 --> 00:35:41,850 Because I can't let the whole semester go 562 00:35:41,850 --> 00:35:46,870 without writing down that, what is it, the grad-- 563 00:35:46,870 --> 00:35:51,920 Is it the curl of a gradient? 564 00:35:51,920 --> 00:36:00,010 The curl of any gradient of u is always zero. 565 00:36:00,010 --> 00:36:01,820 Whatever it is. 566 00:36:01,820 --> 00:36:03,470 Whatever u is. 567 00:36:03,470 --> 00:36:09,550 And this comes from, let me put the other one down 568 00:36:09,550 --> 00:36:14,070 and then I'll just say why. 569 00:36:14,070 --> 00:36:17,070 The other one is like the transpose of this one. 570 00:36:17,070 --> 00:36:20,070 So if I transpose, so this is the zero operator. 571 00:36:20,070 --> 00:36:22,970 Curl times gradient gives the zero. 572 00:36:22,970 --> 00:36:25,670 So if I just transpose I still have zero. 573 00:36:25,670 --> 00:36:27,190 So if it's a transposed gradient, 574 00:36:27,190 --> 00:36:31,190 I have minus divergence, and actually if I transpose curl 575 00:36:31,190 --> 00:36:33,310 I get curl again. 576 00:36:33,310 --> 00:36:39,240 Of any, now I should put in, what should the curl act on? 577 00:36:39,240 --> 00:36:42,550 It acts on a w, I guess is. 578 00:36:42,550 --> 00:36:49,760 No, divergence w, it acts on an S, sorry. 579 00:36:49,760 --> 00:36:54,227 OK, and the minus sign, of course, 580 00:36:54,227 --> 00:36:55,810 isn't going to matter because I've got 581 00:36:55,810 --> 00:36:57,870 a zero on the right-hand side. 582 00:36:57,870 --> 00:37:04,500 So S. Yeah, so if I take any field-- I mean this is like, 583 00:37:04,500 --> 00:37:08,030 real proper vector calculus. 584 00:37:08,030 --> 00:37:13,100 To check these, I call them identities and maybe sometimes 585 00:37:13,100 --> 00:37:16,670 people indicate an identity meaning it's always 586 00:37:16,670 --> 00:37:19,470 true for every u, or for every S. 587 00:37:19,470 --> 00:37:21,640 They'll use the triple equals sign. 588 00:37:21,640 --> 00:37:24,070 Just to say they're really equal. 589 00:37:24,070 --> 00:37:34,580 OK, so we could define the curl, but you've met it elsewhere 590 00:37:34,580 --> 00:37:39,220 and maybe this isn't the time to do that. 591 00:37:39,220 --> 00:37:42,990 What's the key fact, the key little math business 592 00:37:42,990 --> 00:37:45,550 that makes all these true? 593 00:37:45,550 --> 00:37:52,200 So there's sort of a formal math fact that makes them true. 594 00:37:52,200 --> 00:37:55,330 And then there's the physical understanding 595 00:37:55,330 --> 00:38:03,030 of gradients being directions out with no rotation. 596 00:38:03,030 --> 00:38:05,240 So the physical understanding of that. 597 00:38:05,240 --> 00:38:08,100 But the math, the formal math fact 598 00:38:08,100 --> 00:38:11,640 is the fact that the second derivative of u with respect 599 00:38:11,640 --> 00:38:16,660 to x and y is equal to what? 600 00:38:16,660 --> 00:38:21,000 It's equal to second derivative with respect to y and x, yep. 601 00:38:21,000 --> 00:38:25,160 So you would find if you wrote out all the terms here, 602 00:38:25,160 --> 00:38:28,270 or all the terms here, you would find that 603 00:38:28,270 --> 00:38:32,920 just by using that fact, they all cancel each other. 604 00:38:32,920 --> 00:38:34,870 And the book, of course, does that. 605 00:38:34,870 --> 00:38:42,740 So we simply didn't do 3-D in the vector calculus section. 606 00:38:42,740 --> 00:38:45,870 So I'll stop there with that, because it's really 607 00:38:45,870 --> 00:38:50,280 saying that the curl is tremendously important. 608 00:38:50,280 --> 00:38:59,450 It measures vorticity and flow, and it's-- Being able to-- You 609 00:38:59,450 --> 00:39:02,710 know that like, you take the Navier-Stokes equations? 610 00:39:02,710 --> 00:39:09,460 Well, the pressure and the velocity are the, 611 00:39:09,460 --> 00:39:13,260 I'd say primary variables or the natural quantities 612 00:39:13,260 --> 00:39:17,160 to measure, pressure and velocity for a fluid flow. 613 00:39:17,160 --> 00:39:22,410 But you could also use these identities to set up 614 00:39:22,410 --> 00:39:25,590 in terms of other variables. 615 00:39:25,590 --> 00:39:32,890 Just rewrite the equation and you get other things. 616 00:39:32,890 --> 00:39:39,950 Mentioning Navier-Stokes and fluid flow reminds me to say, 617 00:39:39,950 --> 00:39:44,180 we keep using the example of Laplace's equation. 618 00:39:44,180 --> 00:39:48,120 And a person could say wait a minute, get beyond that. 619 00:39:48,120 --> 00:39:49,610 Right? 620 00:39:49,610 --> 00:39:54,050 So why are you always, when you teach finite elements, 621 00:39:54,050 --> 00:39:56,600 why do you always start with Laplace's equation? 622 00:39:56,600 --> 00:40:00,930 OK, well the main reason is it's the simplest one. 623 00:40:00,930 --> 00:40:03,890 It's the one where you can really see what's happening. 624 00:40:03,890 --> 00:40:06,160 More complicated equations would be for, 625 00:40:06,160 --> 00:40:10,300 like, elasticity, plane elasticity, or 3-D elasticity 626 00:40:10,300 --> 00:40:17,040 or other boundary value problems could be quite messy. 627 00:40:17,040 --> 00:40:23,240 But Laplace's equation is not totally a waste of time. 628 00:40:23,240 --> 00:40:26,510 First, it comes up when you have these scalar unknowns. 629 00:40:26,510 --> 00:40:31,370 And then it also comes up in numerical methods 630 00:40:31,370 --> 00:40:33,760 for Navier-Stokes. 631 00:40:33,760 --> 00:40:36,500 So the standard numerical method for Navier-Stokes, 632 00:40:36,500 --> 00:40:42,020 which would come in 18.086, ends up 633 00:40:42,020 --> 00:40:45,600 with Laplace's equation for the pressure. 634 00:40:45,600 --> 00:40:51,910 So to have a fast Laplace solver, as in today's lecture, 635 00:40:51,910 --> 00:40:53,870 pays off. 636 00:40:53,870 --> 00:40:56,400 So I'm just saying Laplace's equation 637 00:40:56,400 --> 00:41:00,020 is important in itself, it has the great advantage 638 00:41:00,020 --> 00:41:02,820 of being the simplest example we could possibly think of. 639 00:41:02,820 --> 00:41:06,460 It's the example where an x+iy trick works. 640 00:41:06,460 --> 00:41:11,960 And it actually comes up in serious big computations, 641 00:41:11,960 --> 00:41:15,040 because the equation for the pressure 642 00:41:15,040 --> 00:41:20,690 comes out to be a Laplace or a Poisson equation. 643 00:41:20,690 --> 00:41:21,290 Now. 644 00:41:21,290 --> 00:41:23,000 I kept going there. 645 00:41:23,000 --> 00:41:23,690 Yeah, thank you. 646 00:41:23,690 --> 00:41:28,970 AUDIENCE: [INAUDIBLE] 647 00:41:28,970 --> 00:41:31,140 PROFESSOR STRANG: We did, as a MATLAB problem. 648 00:41:31,140 --> 00:41:32,390 AUDIENCE: [INAUDIBLE] 649 00:41:32,390 --> 00:41:34,560 PROFESSOR STRANG: Sorry? 650 00:41:34,560 --> 00:41:35,730 And a first order, right. 651 00:41:35,730 --> 00:41:38,560 AUDIENCE: [INAUDIBLE] 652 00:41:38,560 --> 00:41:39,810 PROFESSOR STRANG: Huh. 653 00:41:39,810 --> 00:41:41,580 Yeah. 654 00:41:41,580 --> 00:41:44,351 I would do it the same way but it wouldn't be symmetric, 655 00:41:44,351 --> 00:41:44,850 of course. 656 00:41:44,850 --> 00:41:48,070 That was the point about that convection term, 657 00:41:48,070 --> 00:41:50,650 is-- The diffusion term would be just what we've done, right? 658 00:41:50,650 --> 00:41:53,220 The diffusion term was that second derivative. 659 00:41:53,220 --> 00:41:56,980 And what would it look like in, as long 660 00:41:56,980 --> 00:42:01,340 as we're close to, what would convection-diffusion in 2-D 661 00:42:01,340 --> 00:42:05,230 look like? 662 00:42:05,230 --> 00:42:08,000 Just, I mean part of your interest 663 00:42:08,000 --> 00:42:12,730 is pass 18.085 and get rid of it, right? 664 00:42:12,730 --> 00:42:16,590 But another part is like, these are problems 665 00:42:16,590 --> 00:42:22,197 that if you're in Course 16, Course 2, others, 666 00:42:22,197 --> 00:42:23,280 you're going to meet this. 667 00:42:23,280 --> 00:42:27,430 So the diffusion part is going to be, 668 00:42:27,430 --> 00:42:32,370 again in 2-D I'll have some minus u, well, 669 00:42:32,370 --> 00:42:36,970 I made it simple because I took c(x) to be one. 670 00:42:36,970 --> 00:42:39,460 It could have a c(x) in there. 671 00:42:39,460 --> 00:42:42,360 And what would the convection term look like? 672 00:42:42,360 --> 00:42:49,210 I'd have a velocity, in the x direction, say a V_x. 673 00:42:49,210 --> 00:42:52,820 And a velocity in the y direction, V_y. 674 00:42:52,820 --> 00:42:56,340 V_y times-- That's just a number. 675 00:42:56,340 --> 00:43:01,030 In the simplest case that would be my river is traveling, 676 00:43:01,030 --> 00:43:09,675 or my flow is traveling along, and equals zero. 677 00:43:09,675 --> 00:43:10,550 AUDIENCE: [INAUDIBLE] 678 00:43:10,550 --> 00:43:12,400 PROFESSOR STRANG: Sorry? 679 00:43:12,400 --> 00:43:16,870 Yeah I don't know which way the river's traveling, actually. 680 00:43:16,870 --> 00:43:19,970 So those are just constants. 681 00:43:19,970 --> 00:43:23,070 They could have positive or negative signs. 682 00:43:23,070 --> 00:43:27,800 The V_x and V_y is the constant flow that's carrying, 683 00:43:27,800 --> 00:43:29,050 what am I doing here? 684 00:43:29,050 --> 00:43:32,990 The flow is flowing along, and if those are constants 685 00:43:32,990 --> 00:43:36,300 it's just flowing steady, steady flow. 686 00:43:36,300 --> 00:43:39,840 But it's diffusing at the same time. 687 00:43:39,840 --> 00:43:43,640 And this would bring in that same difficulties 688 00:43:43,640 --> 00:43:47,370 that we met in the MATLAB 1-D. So the MATLAB 1-D problem just 689 00:43:47,370 --> 00:43:50,480 didn't have a y. 690 00:43:50,480 --> 00:43:53,260 I don't care, yeah the sign I'm not worried about, 691 00:43:53,260 --> 00:43:56,490 it's just is that there, now I'm in 2-D. 692 00:43:56,490 --> 00:43:58,380 And what would I expect to see? 693 00:43:58,380 --> 00:44:01,810 I'd expect to see some trouble when V is large. 694 00:44:01,810 --> 00:44:06,960 When V is large, convection, this is convection down here. 695 00:44:06,960 --> 00:44:09,390 This is the convection part. 696 00:44:09,390 --> 00:44:15,800 And if V is large, then so that this should be a lower order 697 00:44:15,800 --> 00:44:19,590 term, is really fighting against this higher order term. 698 00:44:19,590 --> 00:44:24,090 I expect numerical difficulties, just the way we met. 699 00:44:24,090 --> 00:44:30,190 So anyway, if I did a MATLAB example, stretched it to 2-D 700 00:44:30,190 --> 00:44:34,120 we see a whole lot of interesting stuff. 701 00:44:34,120 --> 00:44:37,820 We'd see flow in, flow out, yeah. 702 00:44:37,820 --> 00:44:40,180 But I just can't do everything. 703 00:44:40,180 --> 00:44:42,910 But that would have a weak form, but your question 704 00:44:42,910 --> 00:44:45,150 about weak forms, weak form, when 705 00:44:45,150 --> 00:44:48,380 you have this anti-symmetric term 706 00:44:48,380 --> 00:44:52,870 for odd number of derivatives, is not quite as beautiful. 707 00:44:52,870 --> 00:44:55,050 But you have to deal with it, of course. 708 00:44:55,050 --> 00:44:56,860 Yep. 709 00:44:56,860 --> 00:45:01,810 OK, Ready for whatever. 710 00:45:01,810 --> 00:45:02,980 Any thoughts? 711 00:45:02,980 --> 00:45:05,360 Let's see, just have a look again 712 00:45:05,360 --> 00:45:11,540 at the list of problem topics. 713 00:45:11,540 --> 00:45:14,130 To see if anything occurs to you. 714 00:45:14,130 --> 00:45:16,480 I mean, not that it should. 715 00:45:16,480 --> 00:45:23,210 You know we're OK. 716 00:45:23,210 --> 00:45:27,290 I'm happy to call it a day on that, 717 00:45:27,290 --> 00:45:32,720 and time for dinner for everybody and see you at 7:30.