. 220 00:20:41,000 --> 00:20:46,000 And here, to be completely truthful, I have to say defined 221 00:20:46,000 --> 00:20:50,000 in a simply connected region. Otherwise, we might have the 222 00:20:50,000 --> 00:20:53,000 same kind of strange things happening as before. 223 00:20:53,000 --> 00:21:03,000 Let's not worry too much about it. 224 00:21:03,000 --> 00:21:07,000 For accuracy we need our vector field to be defined in a simply 225 00:21:07,000 --> 00:21:10,000 connected region. And example is just if it is 226 00:21:10,000 --> 00:21:14,000 defined everywhere. If you don't have any evil 227 00:21:14,000 --> 00:21:21,000 eliminators then you can just go ahead and there is no problem. 228 00:21:21,000 --> 00:21:36,000 It is a gradient field. We need three conditions. 229 00:21:36,000 --> 00:21:43,000 Let's do it in order. P sub y equals Q sub x. 230 00:21:43,000 --> 00:21:52,000 And we have P sub z equals R sub x and Q sub z equals R sub 231 00:21:52,000 --> 00:21:55,000 y. How do you remember these three 232 00:21:55,000 --> 00:21:56,000 conditions? Well, it is pretty easy. 233 00:21:56,000 --> 00:22:01,000 You pick any two components, say the x and the z component, 234 00:22:01,000 --> 00:22:04,000 and you take the partial of the x component with respect to z, 235 00:22:04,000 --> 00:22:07,000 the partial of the z component with respect to x and you must 236 00:22:07,000 --> 00:22:12,000 make them equal. And the same with every pair of 237 00:22:12,000 --> 00:22:15,000 variables. In fact, if you had a function 238 00:22:15,000 --> 00:22:17,000 of many more variables the criterion would still look 239 00:22:17,000 --> 00:22:21,000 exactly like that. For every pair of components 240 00:22:21,000 --> 00:22:24,000 the mixed partials must be the same. 241 00:22:24,000 --> 00:22:27,000 But we are not going to go beyond three variables so you 242 00:22:27,000 --> 00:22:32,000 don't need to know that. This you need to know so let me 243 00:22:32,000 --> 00:22:33,000 box it. 244 00:22:51,000 --> 00:23:02,000 That is pretty straightforward. Let's do an example just to see 245 00:23:02,000 --> 00:23:11,000 how it goes. By the way, we can also think 246 00:23:11,000 --> 00:23:19,000 of it in terms of differentials. Before I do the example, 247 00:23:19,000 --> 00:23:22,000 let me just say in a different language. 248 00:23:22,000 --> 00:23:29,000 If we have a differential given to us of a form Pdx Qdy Rdz is 249 00:23:29,000 --> 00:23:33,000 going to be an exact differential, 250 00:23:33,000 --> 00:23:40,000 which means it is equal to df for some function F exactly and 251 00:23:40,000 --> 00:23:43,000 of the same conditions. That is the same thing. 252 00:23:43,000 --> 00:23:51,000 Just in the language of differentials. 253 00:23:51,000 --> 00:23:57,000 The example that I promised. Of course, I could do again the 254 00:23:57,000 --> 00:24:00,000 same one over there and check that it satisfies the condition, 255 00:24:00,000 --> 00:24:02,000 but then it wouldn't be much fun. 256 00:24:02,000 --> 00:24:10,000 So let's do a better one. Actually, let's do it in a way 257 00:24:10,000 --> 00:24:18,000 that looks like an exam problem. Let's say for which a and b is 258 00:24:18,000 --> 00:24:29,000 a xy dx plus -- Oh, it is not going to fit here. 259 00:24:29,000 --> 00:24:39,000 But it will fit here. a xy dx ( x^2 z^3) dy (byz^2 - 260 00:24:39,000 --> 00:24:51,000 4z^3) dz, an exact differential. Or, if you don't like exact 261 00:24:51,000 --> 00:24:54,000 differentials, for which a and b is the 262 00:24:54,000 --> 00:24:58,000 corresponding vector field with i, j and k instead, 263 00:24:58,000 --> 00:25:02,000 a gradient field. Let's just apply the criterion. 264 00:25:02,000 --> 00:25:06,000 And, of course, you can guess that what will 265 00:25:06,000 --> 00:25:11,000 follow is figuring out how to find the potential when there is 266 00:25:11,000 --> 00:25:12,000 one. 267 00:25:33,000 --> 00:25:40,000 Let's do it one by one. We want to compare P sub y with 268 00:25:40,000 --> 00:25:45,000 Q sub x, we want to compare P sub z with 269 00:25:45,000 --> 00:25:53,000 R sub x and we want to compare Q sub z with R sub y where we call 270 00:25:53,000 --> 00:25:57,000 P, Q and R these guys. Let's see. 271 00:25:57,000 --> 00:26:04,000 What is P sub y? That seems to be ax. 272 00:26:04,000 --> 00:26:09,000 What is Q sub x? 2x. 273 00:26:09,000 --> 00:26:12,000 Q is this one. Actually, let me write them 274 00:26:12,000 --> 00:26:16,000 down. Because otherwise I am going to 275 00:26:16,000 --> 00:26:21,000 get confused myself. This guy here, 276 00:26:21,000 --> 00:26:32,000 that is P, this guy here, that is Q and that guy here, 277 00:26:32,000 --> 00:26:38,000 that is R. This one tells us that a should 278 00:26:38,000 --> 00:26:43,000 be equal to two of the first product that you hold. 279 00:26:43,000 --> 00:26:49,000 OK. Let's look at P sub z. That is just zero. 280 00:26:49,000 --> 00:26:51,000 R sub x? Well, R doesn't have any x 281 00:26:51,000 --> 00:26:56,000 either so that is zero. This one is not a problem. 282 00:26:56,000 --> 00:27:02,000 Q sub z? Well, that seems to be 3z2. 283 00:27:02,000 --> 00:27:14,000 R sub y seems to be bz2, so b should be equal to three. 284 00:27:14,000 --> 00:27:19,000 We need to have a equals two and, this is an and, 285 00:27:19,000 --> 00:27:23,000 not or, b equals three for this to be exact. 286 00:27:23,000 --> 00:27:27,000 For those values of a and b, we can look for a potential 287 00:27:27,000 --> 00:27:30,000 using the method that we are going to see right now. 288 00:27:30,000 --> 00:27:33,000 For any other values of a and b we cannot. 289 00:27:33,000 --> 00:27:38,000 If we have to compute a line integral, we have to do it by 290 00:27:38,000 --> 00:27:42,000 finding a parameter and setting up everything. 291 00:27:42,000 --> 00:27:57,000 Any questions at this point? Yes? 292 00:27:57,000 --> 00:28:00,000 I see. Well, if I got the same answer, 293 00:28:00,000 --> 00:28:05,000 oh, did say bz^2 or 3bz^2? Well, 3bz^2, for example, 294 00:28:05,000 --> 00:28:09,000 I would need b to be zero because the only time that 3bz2 295 00:28:09,000 --> 00:28:13,000 equals bz2 as not just at one point but everywhere, 296 00:28:13,000 --> 00:28:15,000 I need them to be the same function of x, 297 00:28:15,000 --> 00:28:18,000 y, z. Well, if a coefficient of z2 is 298 00:28:18,000 --> 00:28:22,000 the same that would be give b equals 3b, that would give me b 299 00:28:22,000 --> 00:28:25,000 equals zero. If you got bz2 on both sides 300 00:28:25,000 --> 00:28:29,000 then it would mean for any value of b it works, 301 00:28:29,000 --> 00:28:35,000 and you wouldn't have to worry about what the value of b is. 302 00:28:35,000 --> 00:28:41,000 Any other questions? No. 303 00:28:41,000 --> 00:28:50,000 OK. Now, how do we find the 304 00:28:50,000 --> 00:28:58,000 potential? Well, there are two methods as 305 00:28:58,000 --> 00:28:59,000 before. One of them, 306 00:28:59,000 --> 00:29:02,000 I don't remember if it was the first one or the second one last 307 00:29:02,000 --> 00:29:04,000 time, but it really doesn't matter. 308 00:29:04,000 --> 00:29:10,000 One of them was just to say that the value of F at the 309 00:29:10,000 --> 00:29:15,000 point, let me call that x1, y1, z1, 310 00:29:15,000 --> 00:29:22,000 is equal to the line integral of my field along a well-chosen 311 00:29:22,000 --> 00:29:25,000 curve plus, of course, a constant, 312 00:29:25,000 --> 00:29:30,000 which is going to be the integration constant. 313 00:29:30,000 --> 00:29:39,000 And the kind of curve that I will take to do this calculation 314 00:29:39,000 --> 00:29:48,000 will just be my favorite curve going from the origin to the 315 00:29:48,000 --> 00:29:51,000 point x1, y1, z1. 316 00:29:51,000 --> 00:29:57,000 And so, typically the most common choice would be to go 317 00:29:57,000 --> 00:30:04,000 just first along the x-axis, then parallel to the y-axis and 318 00:30:04,000 --> 00:30:11,000 then parallel to the z-axis all the way to my point x1, 319 00:30:11,000 --> 00:30:14,000 y1, z1. I would just calculate three 320 00:30:14,000 --> 00:30:18,000 easy line integrals. Add them together and that 321 00:30:18,000 --> 00:30:21,000 would give me the value of my function. 322 00:30:21,000 --> 00:30:27,000 That method works exactly the same way as it did in two 323 00:30:27,000 --> 00:30:30,000 variables. Now, I seem to recall that you 324 00:30:30,000 --> 00:30:32,000 guys mostly preferred the other method. 325 00:30:32,000 --> 00:30:34,000 I am going to tell you about the other method as well, 326 00:30:34,000 --> 00:30:37,000 but I just want to point out this one actually doesn't become 327 00:30:37,000 --> 00:30:40,000 more complicated. The other one has actually more 328 00:30:40,000 --> 00:30:42,000 steps. I mean, of course, 329 00:30:42,000 --> 00:30:45,000 here there are also a bit more steps because you have three 330 00:30:45,000 --> 00:30:47,000 parts to your path instead of two. 331 00:30:47,000 --> 00:30:54,000 You have three line integrals to compute instead of two, 332 00:30:54,000 --> 00:30:59,000 but conceptually it remains exactly the same idea. 333 00:30:59,000 --> 00:31:07,000 I should say it works the same way as in 2D. 334 00:31:07,000 --> 00:31:17,000 Not much changes. Let's look at the other method 335 00:31:17,000 --> 00:31:24,000 using anti-derivatives. Remember we want to find a 336 00:31:24,000 --> 00:31:29,000 function little f whose partials are exactly the things we have 337 00:31:29,000 --> 00:31:31,000 been given. We want to solve, 338 00:31:31,000 --> 00:31:36,000 well, let me plug in the values of a and b that will work. 339 00:31:36,000 --> 00:31:47,000 We said a should be two, so f sub x should be 2xy, 340 00:31:47,000 --> 00:31:59,000 f sub y should be x2 plus z3, and f sub z should be 3yz^2 341 00:31:59,000 --> 00:32:04,000 minus 4z^3. We are going to look at them 342 00:32:04,000 --> 00:32:07,000 one at a time and get partial information on the function. 343 00:32:07,000 --> 00:32:11,000 And then we will compare with the others to get more 344 00:32:11,000 --> 00:32:15,000 information until we are completely done. 345 00:32:15,000 --> 00:32:25,000 The first thing we will do, we know that f sub x is 2xy. 346 00:32:25,000 --> 00:32:27,000 That should tell us something about f. 347 00:32:27,000 --> 00:32:31,000 Well, let's just integrate that with respect to x. 348 00:32:31,000 --> 00:32:36,000 Let me write integral dx next to that. 349 00:32:36,000 --> 00:32:41,000 That tells us that f should be, well, if we integrate that with 350 00:32:41,000 --> 00:32:44,000 respect to x, 2x integrates to x^2, 351 00:32:44,000 --> 00:32:47,000 so we should get x2y. Plus, of course, 352 00:32:47,000 --> 00:32:50,000 an integration constant. Now, what do we mean by 353 00:32:50,000 --> 00:32:53,000 integration constant. It means that for given values 354 00:32:53,000 --> 00:32:58,000 of y and z we will get a term that does not depend on x. 355 00:32:58,000 --> 00:33:02,000 It still depends on y and z. In fact, what we get is a 356 00:33:02,000 --> 00:33:07,000 function of y and z. See, if you took the derivative 357 00:33:07,000 --> 00:33:12,000 of this with respect to x you will get 2xy and this guy will 358 00:33:12,000 --> 00:33:16,000 go away because there is no x in it. 359 00:33:16,000 --> 00:33:19,000 That is the first step. Now we need to get some 360 00:33:19,000 --> 00:33:21,000 information on g. How do we do that? 361 00:33:21,000 --> 00:33:28,000 Well, we look at the other partials. 362 00:33:28,000 --> 00:33:35,000 F sub y, we want that to be x^2 z^3. 363 00:33:35,000 --> 00:33:41,000 But we have another way to find it, which is starting from this 364 00:33:41,000 --> 00:33:50,000 and differentiating. Let me try to use color for 365 00:33:50,000 --> 00:33:54,000 this. Now, if I take the partial of 366 00:33:54,000 --> 00:33:58,000 this with respect to y, I am going to get a different 367 00:33:58,000 --> 00:34:06,000 formula for f sub y. That will be x^2 plus g sub y. 368 00:34:06,000 --> 00:34:15,000 Well, if I compare these two expressions that tells me that g 369 00:34:15,000 --> 00:34:23,000 sub y should be z3. Now, if I have this I can 370 00:34:23,000 --> 00:34:32,000 integrate with respect to y. That will tell me that g is 371 00:34:32,000 --> 00:34:39,000 actually yz^3 plus an integration constant. 372 00:34:39,000 --> 00:34:42,000 That constant, again, does not depend on y, 373 00:34:42,000 --> 00:34:46,000 but it can still depend on z because we still have not said 374 00:34:46,000 --> 00:34:49,000 anything about partial with respect to z. 375 00:34:49,000 --> 00:34:54,000 In fact, that constant I will write as a function h of z. 376 00:34:54,000 --> 00:35:00,000 If I have this function of z and I take its partial with 377 00:35:00,000 --> 00:35:04,000 respect to y, I will still get z^3 no matter 378 00:35:04,000 --> 00:35:06,000 what h was. Now, how do I find h? 379 00:35:06,000 --> 00:35:09,000 Well, obviously, I have to look at f sub z. 380 00:35:41,000 --> 00:35:47,000 F sub z. We know from the given vector 381 00:35:47,000 --> 00:35:53,000 field that we want it to be 3yz^2 minus 4z^3. 382 00:35:53,000 --> 00:35:56,000 In case you are wondering where that came from, 383 00:35:56,000 --> 00:36:01,000 that was R. But that is also obtained by 384 00:36:01,000 --> 00:36:09,000 differentiating with respect to z what we had so far. 385 00:36:09,000 --> 00:36:15,000 Sorry. What did we have so far? 386 00:36:15,000 --> 00:36:20,000 Well, we had f equals x^2y plus g. 387 00:36:20,000 --> 00:36:30,000 And we said g is actually yz^3 plus h of z. 388 00:36:30,000 --> 00:36:37,000 That is what we have so far. If we take the derivative of 389 00:36:37,000 --> 00:36:44,000 that with respect to z, we will get zero plus 3yz^2 390 00:36:44,000 --> 00:36:51,000 plus h prime of z, or dh dz as you want. 391 00:36:51,000 --> 00:36:59,000 Now, if we compare these two, we will get the derivative of 392 00:36:59,000 --> 00:37:03,000 h. It will tell us that h prime is 393 00:37:03,000 --> 00:37:08,000 negative for z3. That means that h is negative 394 00:37:08,000 --> 00:37:13,000 z^4 plus a constant. And this it is at last an 395 00:37:13,000 --> 00:37:17,000 actual constant. Because it does not depend on z 396 00:37:17,000 --> 00:37:21,000 and there is nothing else for it to depend on. 397 00:37:21,000 --> 00:37:33,000 Now we plug this into what we had before, and that will give 398 00:37:33,000 --> 00:37:44,000 us our function f. We get that f=x^2y yz^3 - z^4 399 00:37:44,000 --> 00:37:49,000 plus constant. If you just wanted to find one 400 00:37:49,000 --> 00:37:51,000 potential, you can just forget the constant. 401 00:37:51,000 --> 00:37:55,000 This guy was a potential. If you want all the potentials 402 00:37:55,000 --> 00:37:58,000 they differ by this constant. OK. 403 00:37:58,000 --> 00:38:01,000 Just to recap the method what did we do? 404 00:38:01,000 --> 00:38:04,000 We started with -- And, of course, you can do it in 405 00:38:04,000 --> 00:38:07,000 whichever order you prefer, but you have to still follow 406 00:38:07,000 --> 00:38:11,000 the systematic method. You start with f sub x and you 407 00:38:11,000 --> 00:38:13,000 integrate that with respect to x. 408 00:38:13,000 --> 00:38:19,000 That gives you f up to a function of y and z only. 409 00:38:19,000 --> 00:38:25,000 Now you compare f sub y as given to you by the vector field 410 00:38:25,000 --> 00:38:32,000 with the formula you get from this expression for f. 411 00:38:32,000 --> 00:38:36,000 And, of course, this one will involve g sub y. 412 00:38:36,000 --> 00:38:41,000 Out of this, you will get the value of g sub 413 00:38:41,000 --> 00:38:44,000 y. When you have g sub y that 414 00:38:44,000 --> 00:38:48,000 gives you g up to a function of z only. 415 00:38:48,000 --> 00:38:52,000 And so now you have f up to a function of z only. 416 00:38:52,000 --> 00:38:57,000 And what you will do is look at the derivative with respect to 417 00:38:57,000 --> 00:38:59,000 z, the one you want coming from 418 00:38:59,000 --> 00:39:02,000 the vector field and the one you have coming from this formula 419 00:39:02,000 --> 00:39:05,000 for f, match them and that will tell 420 00:39:05,000 --> 00:39:09,000 you h prime. You will get h and then you 421 00:39:09,000 --> 00:39:20,000 will get f. Any questions? 422 00:39:20,000 --> 00:39:25,000 Who still prefers this method? OK, still most of you. 423 00:39:25,000 --> 00:39:29,000 Who is thinking that maybe the other method was not so bad 424 00:39:29,000 --> 00:39:33,000 after all? OK. That is still a minority. 425 00:39:33,000 --> 00:39:36,000 You can choose whichever one you prefer. 426 00:39:36,000 --> 00:39:42,000 I would encourage you to get some practice by trying both on 427 00:39:42,000 --> 00:39:47,000 least a couple of examples just to make sure that you know how 428 00:39:47,000 --> 00:39:53,000 to do them both and then stick to whichever one you prefer. 429 00:39:53,000 --> 00:39:59,000 Any questions on that? No. I guess I already asked. 430 00:39:59,000 --> 00:40:07,000 Still no questions? OK. 431 00:40:07,000 --> 00:40:13,000 The next logical thing is going to be curl. 432 00:40:13,000 --> 00:40:17,000 And the theorem that is going to replace Green's theorem for 433 00:40:17,000 --> 00:40:21,000 work in this setting is going to be called Stokes' theorem. 434 00:40:21,000 --> 00:40:34,000 Let me start by telling you about curl in 3D. 435 00:40:34,000 --> 00:40:39,000 Here is the statement. The curl is just going to 436 00:40:39,000 --> 00:40:43,000 measure how much your vector field fails to be conservative. 437 00:40:43,000 --> 00:40:47,000 And, if you want to think about it in terms of motions, 438 00:40:47,000 --> 00:40:51,000 that also will measure the rotation part of the motion. 439 00:40:51,000 --> 00:40:55,000 Well, let me first give a definition. 440 00:40:55,000 --> 00:41:00,000 Let's say that my vector field has components P, 441 00:41:00,000 --> 00:41:06,000 Q and R. Then we define the curl of F to 442 00:41:06,000 --> 00:41:16,000 be R sub y minus Q sub z times i plus P sub z minus R sub x times 443 00:41:16,000 --> 00:41:21,000 j plus Q sub x minus P sub y times k. 444 00:41:21,000 --> 00:41:25,000 And of course nobody can remember this formula, 445 00:41:25,000 --> 00:41:28,000 so what is the structure of this formula? 446 00:41:28,000 --> 00:41:35,000 Well, you see, each of these guys is one of 447 00:41:35,000 --> 00:41:43,000 the things that have to be zero for our field to be 448 00:41:43,000 --> 00:41:52,000 conservative. If F is defined in a simply 449 00:41:52,000 --> 00:42:05,000 connected region then we have that F is conservative and is 450 00:42:05,000 --> 00:42:15,000 equivalent to if and only if curl F is zero. 451 00:42:15,000 --> 00:42:19,000 Now, an important difference between curl here and curl in 452 00:42:19,000 --> 00:42:23,000 the plane is that now the curl of a vector field is again a 453 00:42:23,000 --> 00:42:27,000 vector field. These expressions are functions 454 00:42:27,000 --> 00:42:31,000 of x, y, z and together you form a vector out of them. 455 00:42:31,000 --> 00:42:36,000 The curl of a vector field in space is actually a vector 456 00:42:36,000 --> 00:42:43,000 field, not a scalar function. I have delayed the inevitable. 457 00:42:43,000 --> 00:42:47,000 I have to really tell you how to remember this evil formula. 458 00:42:47,000 --> 00:42:55,000 The secret is that, in fact, you can think of this 459 00:42:55,000 --> 00:43:01,000 as del cross f. Maybe you have seen that in 460 00:43:01,000 --> 00:43:04,000 physics. This is really where this del 461 00:43:04,000 --> 00:43:08,000 notation becomes extremely useful, because that is 462 00:43:08,000 --> 00:43:13,000 basically the only way to remember the formula for curl. 463 00:43:30,000 --> 00:43:34,000 Remember we introduced the dell operator. 464 00:43:34,000 --> 00:43:42,000 That was this symbolic vector operator in which the components 465 00:43:42,000 --> 00:43:47,000 are the partial derivative operators. 466 00:43:47,000 --> 00:43:59,000 We have seen that if you apply this to a scalar function then 467 00:43:59,000 --> 00:44:08,000 that will give you the gradient. And we have seen that if you do 468 00:44:08,000 --> 00:44:13,000 the dot product between dell and a vector field, 469 00:44:13,000 --> 00:44:19,000 maybe I should give it components P, 470 00:44:19,000 --> 00:44:24,000 Q and R, you will get partial P over 471 00:44:24,000 --> 00:44:30,000 partial x plus partial Q over partial y plus partial R over 472 00:44:30,000 --> 00:44:36,000 partial z, which is the divergence. 473 00:44:36,000 --> 00:44:47,000 And so now what is new is that if I try to do dell cross F, 474 00:44:47,000 --> 00:44:53,000 well, what is dell cross F? I have to set up a 475 00:44:53,000 --> 00:44:58,000 cross-product between this strange thing that is not really 476 00:44:58,000 --> 00:45:02,000 a vector. I mean, I cannot really think 477 00:45:02,000 --> 00:45:05,000 of partial over partial x as a number. 478 00:45:05,000 --> 00:45:10,000 And my vector field

. 479 00:45:10,000 --> 00:45:14,000 See, that is really a completely perverted use of a 480 00:45:14,000 --> 00:45:18,000 determinant notation. Initially, determinants were 481 00:45:18,000 --> 00:45:21,000 just supposed to be you had a three by three table of numbers 482 00:45:21,000 --> 00:45:23,000 and you computed a number out of them. 483 00:45:23,000 --> 00:45:28,000 These guys are functions so they count as numbers, 484 00:45:28,000 --> 00:45:33,000 but these are vectors and these are partial derivatives. 485 00:45:33,000 --> 00:45:37,000 It doesn't really make much sense, except this notation. 486 00:45:37,000 --> 00:45:43,000 If you try to enter this into a calculator or computer, 487 00:45:43,000 --> 00:45:48,000 it will just yell back at you saying are you crazy. 488 00:45:48,000 --> 00:45:50,000 [LAUGHTER] We just use that as a notation 489 00:45:50,000 --> 00:45:53,000 to remember what is in there. Let's try and see how that 490 00:45:53,000 --> 00:45:55,000 works. The component of i in this 491 00:45:55,000 --> 00:45:58,000 cross-product, remember that is this smaller 492 00:45:58,000 --> 00:46:01,000 determinant, that smaller determinant is 493 00:46:01,000 --> 00:46:07,000 partial over partial y of R minus partial over partial z of 494 00:46:07,000 --> 00:46:10,000 Q, the coefficient of i. 495 00:46:10,000 --> 00:46:14,000 And that seems to be what I had over there. 496 00:46:14,000 --> 00:46:18,000 If not then I made a mistake. Minus the next determinant 497 00:46:18,000 --> 00:46:20,000 times z. Remember there is always a 498 00:46:20,000 --> 00:46:23,000 minus sign in front of a j component when you do a 499 00:46:23,000 --> 00:46:27,000 cross-product. The other one is partial over 500 00:46:27,000 --> 00:46:33,000 partial x R minus partial over partial z of P plus the 501 00:46:33,000 --> 00:46:39,000 component of z which is going to be partial over partial x Q 502 00:46:39,000 --> 00:46:46,000 minus partial over partial y P. And that is indeed going to be 503 00:46:46,000 --> 00:46:48,000 the curl of F. In practice, 504 00:46:48,000 --> 00:46:51,000 if you have to compute the curl of a vector field, 505 00:46:51,000 --> 00:46:53,000 you know, don't try to remember this formula. 506 00:46:53,000 --> 00:46:59,000 Just set up this cross-product with whatever formulas you have 507 00:46:59,000 --> 00:47:04,000 for the components of a field and then compute it. 508 00:47:04,000 --> 00:47:15,000 Don't bother to try to remember the general formula, 509 00:47:15,000 --> 00:47:25,000 just remember this. What is the geometric 510 00:47:25,000 --> 00:47:35,000 interpretation of curl, just to finish? 511 00:47:35,000 --> 00:47:48,000 In a way, I will say just curl measures the rotation component 512 00:47:48,000 --> 00:47:55,000 in a velocity field. An exercise that you can do, 513 00:47:55,000 --> 00:47:59,000 which is actually pretty easy to check, is say that we have a 514 00:47:59,000 --> 00:48:04,000 fluid that is just rotating about the x-axis uniformly. 515 00:48:04,000 --> 00:48:10,000 Your fluid is just rotating like that about the z-axis. 516 00:48:10,000 --> 00:48:15,000 If I take a rotation about the z-axis. 517 00:48:15,000 --> 00:48:28,000 That is given by a velocity field with components at angular 518 00:48:28,000 --> 00:48:35,000 velocity omega. That will be negative omega 519 00:48:35,000 --> 00:48:41,000 times y, then omega x and zero. And the curl of that you can 520 00:48:41,000 --> 00:48:45,000 compute, and you will find two omega times k. 521 00:48:45,000 --> 00:48:48,000 Concretely, this curl gives you the angular 522 00:48:48,000 --> 00:48:51,000 velocity of the rotation, well, with a factor two but 523 00:48:51,000 --> 00:48:53,000 that doesn't matter, and the axis of rotation, 524 00:48:53,000 --> 00:48:56,000 the direction of the axis of rotation. 525 00:48:56,000 --> 00:48:59,000 It tells you it is rotating about a vertical axis. 526 00:48:59,000 --> 00:49:01,000 And, in general, if you have a complicated 527 00:49:01,000 --> 00:49:03,000 motion some of it might be, you know, there is a 528 00:49:03,000 --> 00:49:05,000 translation. And then within that 529 00:49:05,000 --> 00:49:08,000 translation there is maybe expansion and rotation and 530 00:49:08,000 --> 00:49:11,000 sharing and everything. And the curl will compute how 531 00:49:11,000 --> 00:49:14,000 much rotation is taking place. It will tell you, 532 00:49:14,000 --> 00:49:16,000 say that you have a very small solid, 533 00:49:16,000 --> 00:49:19,000 I don't know like a ping pong ball in your flow, 534 00:49:19,000 --> 00:49:21,000 and it is just going with the flow, 535 00:49:21,000 --> 00:49:25,000 it tells you how it is going to start rotating. 536 00:49:25,000 --> 00:49:32,000 That is what curl measures. On Thursday we will see Stokes' 537 00:49:32,000 --> 00:49:37,000 theorem, which will be the last ingredient before the next exam. 538 00:49:37,000 --> 00:49:40,000 And then on Friday we will review stuff.