WEBVTT 00:00:01.000 --> 00:00:03.000 The following content is provided under a Creative 00:00:03.000 --> 00:00:05.000 Commons license. Your support will help MIT 00:00:05.000 --> 00:00:08.000 OpenCourseWare continue to offer high quality educational 00:00:08.000 --> 00:00:13.000 resources for free. To make a donation or to view 00:00:13.000 --> 00:00:18.000 additional materials from hundreds of MIT courses, 00:00:18.000 --> 00:00:23.000 visit MIT OpenCourseWare at ocw.mit.edu. 00:00:23.000 --> 00:00:25.000 And, well let's see. So, before we actually start 00:00:25.000 --> 00:00:30.000 reviewing for the test, I still have to tell you a few 00:00:30.000 --> 00:00:34.000 small things because I promised to say a few words about what's 00:00:34.000 --> 00:00:37.000 the difference, or precisely, 00:00:37.000 --> 00:00:41.000 what's the difference between curl being zero and a field 00:00:41.000 --> 00:00:45.000 being a gradient field, and why we have this assumption 00:00:45.000 --> 00:00:49.000 that our vector field had to be defined everywhere for a field 00:00:49.000 --> 00:00:53.000 with curl zero to actually be conservative for our test for 00:00:53.000 --> 00:01:04.000 gradient fields to be valid? So -- More about validity of 00:01:04.000 --> 00:01:17.000 Green's theorem and things like that. 00:01:17.000 --> 00:01:29.000 So, we've seen the statement of Green's theorem in two forms. 00:01:29.000 --> 00:01:33.000 Both of them have to do with comparing a line integral along 00:01:33.000 --> 00:01:38.000 a closed curve to a double integral over the region inside 00:01:38.000 --> 00:01:41.000 enclosed by the curve. So, 00:01:41.000 --> 00:01:45.000 one of them says the line integral for the work done by a 00:01:45.000 --> 00:01:50.000 vector field along a closed curve counterclockwise is equal 00:01:50.000 --> 00:01:54.000 to the double integral of a curl of a field over the enclosed 00:01:54.000 --> 00:01:58.000 region. And, the other one says the 00:01:58.000 --> 00:02:05.000 total flux out of the region, so, the flux through the curve 00:02:05.000 --> 00:02:11.000 is equal to the double integral of divergence of a field in the 00:02:11.000 --> 00:02:13.000 region. So, in both cases, 00:02:13.000 --> 00:02:18.000 we need the vector field to be defined not only, 00:02:18.000 --> 00:02:21.000 I mean, the left hand side makes sense if a vector field is 00:02:21.000 --> 00:02:24.000 just defined on the curve because it's just a line 00:02:24.000 --> 00:02:27.000 integral on C. We don't care what happens 00:02:27.000 --> 00:02:29.000 inside. But, for the right-hand side to 00:02:29.000 --> 00:02:31.000 make sense, and therefore for the equality 00:02:31.000 --> 00:02:34.000 to make sense, we need the vector field to be 00:02:34.000 --> 00:02:38.000 defined everywhere inside the region. 00:02:38.000 --> 00:02:41.000 So, I said, if there is a point somewhere in here where my 00:02:41.000 --> 00:02:45.000 vector field is not defined, then it doesn't work. 00:02:45.000 --> 00:02:53.000 And actually, we've seen that example. 00:02:53.000 --> 00:03:08.000 So, this only works if F and its derivatives are defined 00:03:08.000 --> 00:03:17.000 everywhere in the region, R. 00:03:17.000 --> 00:03:20.000 Otherwise, we are in trouble. OK, 00:03:20.000 --> 00:03:29.000 so we've seen for example that if I gave you the vector field 00:03:29.000 --> 00:03:35.000 minus yi xj over x squared plus y squared, 00:03:35.000 --> 00:03:40.000 so that's the same vector field that was on that problem set a 00:03:40.000 --> 00:03:45.000 couple of weeks ago. Then, well, f is not defined at 00:03:45.000 --> 00:03:52.000 the origin, but it's defined everywhere else. 00:03:52.000 --> 00:04:03.000 And, wherever it's defined, it's curl is zero. 00:04:03.000 --> 00:04:14.000 I should say everywhere it's -- And so, if we have a closed 00:04:14.000 --> 00:04:24.000 curve in the plane, well, there's two situations. 00:04:24.000 --> 00:04:28.000 One is if it does not enclose the origin. 00:04:28.000 --> 00:04:30.000 Then, yes, 00:04:30.000 --> 00:04:36.000 we can apply Green's theorem and it will tell us that it's 00:04:36.000 --> 00:04:42.000 equal to the double integral in here of curl F dA, 00:04:42.000 --> 00:04:46.000 which will be zero because this is zero. 00:04:46.000 --> 00:04:51.000 However, if I have a curve that encloses 00:04:51.000 --> 00:04:54.000 the origin, let's say like this, 00:04:54.000 --> 00:05:03.000 for example, then, 00:05:03.000 --> 00:05:07.000 well, I cannot use the same method because the vector field 00:05:07.000 --> 00:05:11.000 and its curl are not defined at the origin. 00:05:11.000 --> 00:05:14.000 And, in fact, you know that ignoring the 00:05:14.000 --> 00:05:17.000 problem and saying, well, the curl is still zero 00:05:17.000 --> 00:05:19.000 everywhere, will give you the wrong answer 00:05:19.000 --> 00:05:23.000 because we've seen an example. We've seen that along the unit 00:05:23.000 --> 00:05:27.000 circle the total work is 2 pi not zero. 00:05:27.000 --> 00:05:35.000 So, we can't use Green. However, we can't use it 00:05:35.000 --> 00:05:38.000 directly. So, there is an extended 00:05:38.000 --> 00:05:44.000 version of Green's theorem that tells you the following thing. 00:05:44.000 --> 00:05:49.000 Well, it tells me that even though I 00:05:49.000 --> 00:05:54.000 can't do things for just this region enclosed by C prime, 00:05:54.000 --> 00:05:58.000 I can still do things for the region in between two different 00:05:58.000 --> 00:06:06.000 curves. OK, so let me show you what I 00:06:06.000 --> 00:06:11.000 have in mind. So, let's say that I have my 00:06:11.000 --> 00:06:15.000 curve C'. Where's my yellow chalk? 00:06:15.000 --> 00:06:22.000 Oh, here. So, I have this curve C'. 00:06:22.000 --> 00:06:30.000 I can't apply Green's theorem inside it, but let's get out the 00:06:30.000 --> 00:06:34.000 smaller thing. So, that one I'm going to make 00:06:34.000 --> 00:06:41.000 going clockwise. You will see why. 00:06:41.000 --> 00:06:48.000 Then, I could say, well, let me change my mind. 00:06:48.000 --> 00:06:49.000 This picture is not very well prepared. 00:06:49.000 --> 00:06:53.000 That's because my writer is on strike. 00:06:53.000 --> 00:07:02.000 OK, so let's say we have C' and C'' both going counterclockwise. 00:07:02.000 --> 00:07:05.000 Then, I claim that Green's theorem 00:07:05.000 --> 00:07:09.000 still applies, and tells me that the line 00:07:09.000 --> 00:07:14.000 integral along C prime minus the line integral along C double 00:07:14.000 --> 00:07:19.000 prime is equal to the double integral over the region in 00:07:19.000 --> 00:07:23.000 between. So here, now, 00:07:23.000 --> 00:07:35.000 it's this region with the hole of the curve. 00:07:35.000 --> 00:07:41.000 And, well, in our case, that will turn out to be zero 00:07:41.000 --> 00:07:45.000 because curl is zero. OK, so this doesn't tell us 00:07:45.000 --> 00:07:47.000 what each of these two line integrals is. 00:07:47.000 --> 00:07:49.000 But actually, it tells us that they are equal 00:07:49.000 --> 00:07:51.000 to each other. And so, by computing one, 00:07:51.000 --> 00:07:53.000 you can see actually that for this vector field, 00:07:53.000 --> 00:07:57.000 if you take any curve that goes counterclockwise around the 00:07:57.000 --> 00:08:00.000 origin, you would get two pi no matter 00:08:00.000 --> 00:08:04.000 what the curve is. So how do you get to this? 00:08:04.000 --> 00:08:07.000 Why is this not like conceptually a new theorem? 00:08:07.000 --> 00:08:13.000 Well, just think of the following thing. 00:08:13.000 --> 00:08:17.000 I'm not going to do it on top of that because it's going to be 00:08:17.000 --> 00:08:23.000 messy if I draw too many things. But, so here I have my C''. 00:08:23.000 --> 00:08:29.000 Here, I have C'. Let me actually make a slit 00:08:29.000 --> 00:08:33.000 that will connect them to each other like this. 00:08:33.000 --> 00:08:37.000 So now if I take, see, 00:08:37.000 --> 00:08:43.000 I can form a single closed curve that will enclose all of 00:08:43.000 --> 00:08:49.000 this region with kind of an infinitely thin slit here 00:08:49.000 --> 00:08:52.000 counterclockwise. And so, if I go 00:08:52.000 --> 00:08:55.000 counterclockwise around this region, basically I go 00:08:55.000 --> 00:08:58.000 counterclockwise along the outer curve. 00:08:58.000 --> 00:09:01.000 Then I go along the slit. Then I go clockwise along the 00:09:01.000 --> 00:09:04.000 inside curve, then back along the slit. 00:09:04.000 --> 00:09:07.000 And then I'm done. So, 00:09:07.000 --> 00:09:11.000 if I take the line integral along this big curve consisting 00:09:11.000 --> 00:09:15.000 of all these pieces, now I can apply Green's theorem 00:09:15.000 --> 00:09:19.000 to that because it is the usual counterclockwise curve that goes 00:09:19.000 --> 00:09:22.000 around a region where my field is well-defined. 00:09:22.000 --> 00:09:27.000 See, I've eliminated the origin from the picture. 00:09:27.000 --> 00:09:37.000 And, so the total line integral for this thing is equal to the 00:09:37.000 --> 00:09:46.000 integral along C prime, I guess the outer one. 00:09:46.000 --> 00:09:50.000 Then, I also need to have what I do along the inner side. 00:09:50.000 --> 00:09:52.000 And, the inner side is going to be C double prime, 00:09:52.000 --> 00:09:57.000 but going backwards because now I'm going clockwise on C prime 00:09:57.000 --> 00:10:01.000 so that I'm going counterclockwise around the 00:10:01.000 --> 00:10:04.000 shaded region. Well, of course there will be 00:10:04.000 --> 00:10:06.000 contributions from the line integral along this wide 00:10:06.000 --> 00:10:08.000 segment. But, I do it twice, 00:10:08.000 --> 00:10:17.000 once each way. So, they cancel out. 00:10:17.000 --> 00:10:21.000 So, the white segments cancel out. 00:10:21.000 --> 00:10:23.000 You probably shouldn't, in your notes, 00:10:23.000 --> 00:10:25.000 write down white segments because probably they are not 00:10:25.000 --> 00:10:29.000 white on your paper. But, hopefully you get the 00:10:29.000 --> 00:10:33.000 meaning of what I'm trying to say. 00:10:33.000 --> 00:10:36.000 OK, so basically that tells you, you can still play tricks 00:10:36.000 --> 00:10:39.000 with Green's theorem when the region has holes in it. 00:10:39.000 --> 00:10:44.000 You just had to be careful and somehow subtract some other 00:10:44.000 --> 00:10:48.000 curve so that together things will work out. 00:10:48.000 --> 00:10:51.000 There is a similar thing with the divergence theorem, 00:10:51.000 --> 00:10:55.000 of course, with flux and double integral of div f, 00:10:55.000 --> 00:10:58.000 you can apply exactly the same argument. 00:10:58.000 --> 00:11:02.000 OK, so basically you can apply Green's theorem for a region 00:11:02.000 --> 00:11:04.000 that has several boundary curves. 00:11:04.000 --> 00:11:07.000 You just have to be careful that the outer boundary must go 00:11:07.000 --> 00:11:13.000 counterclockwise. The inner boundary either goes 00:11:13.000 --> 00:11:19.000 clockwise, or you put a minus sign. 00:11:19.000 --> 00:11:26.000 OK, and the last cultural note, 00:11:26.000 --> 00:11:34.000 so, the definition, we say that a region in the 00:11:34.000 --> 00:11:36.000 plane, sorry, I should say a connected 00:11:36.000 --> 00:11:45.000 region in the plane, so that means -- So, 00:11:45.000 --> 00:11:47.000 connected means it consists of a single piece. 00:11:47.000 --> 00:11:50.000 OK, so, connected, there is a single piece. 00:11:50.000 --> 00:11:53.000 These two guys together are not connected. 00:11:53.000 --> 00:11:58.000 But, if I join them, then this is a connected 00:11:58.000 --> 00:12:08.000 region. We say it's simply connected -- 00:12:08.000 --> 00:12:17.000 -- if any closed curve in it, OK, 00:12:17.000 --> 00:12:18.000 so I need to gave a name to my region, 00:12:18.000 --> 00:12:22.000 let's say R, any closed curve in R, 00:12:22.000 --> 00:12:29.000 bounds, no, 00:12:29.000 --> 00:12:37.000 sorry. If the interior of any closed 00:12:37.000 --> 00:12:49.000 curve in R -- -- is also contained in R. 00:12:49.000 --> 00:12:51.000 So, concretely, what does that mean? 00:12:51.000 --> 00:12:57.000 That means the region, R, does not have any holes 00:12:57.000 --> 00:13:02.000 inside it. Maybe I should draw two 00:13:02.000 --> 00:13:08.000 pictures to explain what I mean. So, 00:13:08.000 --> 00:13:17.000 this guy here is simply connected while -- -- this guy 00:13:17.000 --> 00:13:29.000 here is not simply connected because if I take this curve, 00:13:29.000 --> 00:13:34.000 that's a curve inside my region. But, the piece that it bounds 00:13:34.000 --> 00:13:38.000 is not actually entirely contained in my origin. 00:13:38.000 --> 00:13:41.000 And, so why is that relevant? Well, 00:13:41.000 --> 00:13:45.000 if you know that your vector field is defined everywhere in a 00:13:45.000 --> 00:13:47.000 simply connected region, then you don't have to worry 00:13:47.000 --> 00:13:50.000 about this question of, can I apply Green's theorem to 00:13:50.000 --> 00:13:52.000 the inside? You know it's automatically OK 00:13:52.000 --> 00:13:54.000 because if you have a closed curve, 00:13:54.000 --> 00:13:59.000 then the vector field is, I mean, if a vector field is 00:13:59.000 --> 00:14:03.000 defined on the curve it will also be defined inside. 00:14:03.000 --> 00:14:11.000 OK, so if the domain of definition 00:14:11.000 --> 00:14:25.000 -- -- of a vector field is defined and differentiable -- -- 00:14:25.000 --> 00:14:38.000 is simply connected -- -- then we can always apply -- -- 00:14:38.000 --> 00:14:47.000 Green's theorem -- -- and, of course, 00:14:47.000 --> 00:14:49.000 provided that we do it on a curve where the vector field is 00:14:49.000 --> 00:14:50.000 defined. I mean, your line integral 00:14:50.000 --> 00:14:53.000 doesn't make sense so there's nothing to compute. 00:14:53.000 --> 00:14:56.000 But, if you have, so, again, the argument would 00:14:56.000 --> 00:14:59.000 be, well, if a vector field is defined on the curve, 00:14:59.000 --> 00:15:01.000 it's also defined inside. So, 00:15:01.000 --> 00:15:04.000 see, the problem with that vector 00:15:04.000 --> 00:15:07.000 field here is precisely that its domain of definition is not 00:15:07.000 --> 00:15:09.000 simply connected because there is a hole, 00:15:09.000 --> 00:15:17.000 namely the origin. OK, so for this guy, 00:15:17.000 --> 00:15:28.000 domain of definition, which is plane minus the origin 00:15:28.000 --> 00:15:39.000 with the origin removed is not simply connected. 00:15:39.000 --> 00:15:42.000 And so that's why you have this line integral that makes perfect 00:15:42.000 --> 00:15:45.000 sense, but you can't apply Green's theorem to it. 00:15:45.000 --> 00:15:47.000 So now, what does that mean a particular? 00:15:47.000 --> 00:15:51.000 Well, we've seen this criterion that if a curl of the vector 00:15:51.000 --> 00:15:55.000 field is zero and it's defined in the entire plane, 00:15:55.000 --> 00:15:58.000 then the vector field is conservative, 00:15:58.000 --> 00:16:01.000 and it's a gradient field. And, the argument to prove that 00:16:01.000 --> 00:16:03.000 is basically to use Green's theorem. 00:16:03.000 --> 00:16:07.000 So, in fact, the actual optimal statement 00:16:07.000 --> 00:16:11.000 you can make is if a vector field is defined in a simply 00:16:11.000 --> 00:16:13.000 connected region, and its curl is zero, 00:16:13.000 --> 00:16:26.000 then it's a gradient field. So, let me just write that down. 00:16:26.000 --> 00:16:29.000 So, the correct statement, I mean, the previous one we've 00:16:29.000 --> 00:16:35.000 seen is also correct. But this one is somehow better 00:16:35.000 --> 00:16:45.000 and closer to what exactly is needed if curl F is zero and the 00:16:45.000 --> 00:16:55.000 domain of definition where F is defined is simply connected -- 00:16:55.000 --> 00:17:04.000 -- then F is conservative. And that means also it's a 00:17:04.000 --> 00:17:11.000 gradient field. It's the same thing. 00:17:11.000 --> 00:17:23.000 OK, any questions on this? No? 00:17:23.000 --> 00:17:27.000 OK, some good news. What I've just said here won't 00:17:27.000 --> 00:17:31.000 come up on the test on Thursday. OK. 00:17:31.000 --> 00:17:35.000 (APPLAUSE) Still, it's stuff that you should be 00:17:35.000 --> 00:17:39.000 aware of generally speaking because it will be useful, 00:17:39.000 --> 00:17:42.000 say, on the next week's problem set. 00:17:42.000 --> 00:17:46.000 And, maybe on the final it would be, 00:17:46.000 --> 00:17:48.000 there won't be any really, really complicated things 00:17:48.000 --> 00:17:53.000 probably, but you might need to be at 00:17:53.000 --> 00:18:01.000 least vaguely aware of this issue of things being simply 00:18:01.000 --> 00:18:04.000 connected. And by the way, 00:18:04.000 --> 00:18:08.000 I mean, this is also somehow the starting point of topology, 00:18:08.000 --> 00:18:12.000 which is the branch of math that studies the shapes of 00:18:12.000 --> 00:18:13.000 regions. So, 00:18:13.000 --> 00:18:15.000 in particular, you can try to distinguish 00:18:15.000 --> 00:18:18.000 domains in the plains by looking at whether they're simply 00:18:18.000 --> 00:18:21.000 connected or not, and what kinds of features they 00:18:21.000 --> 00:18:25.000 have in terms of how you can joint point what kinds of curves 00:18:25.000 --> 00:18:28.000 exist in them. And, since that's the branch of 00:18:28.000 --> 00:18:32.000 math in which I work, I thought I should tell you a 00:18:32.000 --> 00:18:41.000 bit about it. OK, so now back to reviewing 00:18:41.000 --> 00:18:47.000 for the exam. So, I'm going to basically list 00:18:47.000 --> 00:18:49.000 topics. And, if time permits, 00:18:49.000 --> 00:18:53.000 I will say a few things about problems from practice exam 3B. 00:18:53.000 --> 00:18:56.000 I'm hoping that you have it or your neighbor has it, 00:18:56.000 --> 00:18:59.000 or you can somehow get it. Anyway, given time, 00:18:59.000 --> 00:19:04.000 I'm not sure how much I will say about the problems in and of 00:19:04.000 --> 00:19:08.000 themselves. OK, so the main thing to know 00:19:08.000 --> 00:19:13.000 about this exam is how to set up and evaluate double integrals 00:19:13.000 --> 00:19:17.000 and line integrals. OK, if you know how to do these 00:19:17.000 --> 00:19:20.000 two things, then you are in much better shape than if you don't. 00:19:26.000 --> 00:19:43.000 And -- So, the first thing we've seen, just to write it 00:19:43.000 --> 00:19:55.000 down, there's two main objects. And, it's kind of important to 00:19:55.000 --> 00:19:57.000 not confuse them with each other. 00:19:57.000 --> 00:20:02.000 OK, there's double integrals of our regions of some quantity, 00:20:02.000 --> 00:20:06.000 dA, and the other one is the line 00:20:06.000 --> 00:20:11.000 integral along a curve of a vector field, 00:20:11.000 --> 00:20:17.000 F.dr or F.Mds depending on whether it's work or flux that 00:20:17.000 --> 00:20:21.000 we are trying to do. And, so we should know how to 00:20:21.000 --> 00:20:24.000 set up these things and how to evaluate them. 00:20:24.000 --> 00:20:27.000 And, roughly speaking, in this one you start by 00:20:27.000 --> 00:20:32.000 drawing a picture of the region, then deciding which way you 00:20:32.000 --> 00:20:34.000 will integrate it. It could be dx dy, 00:20:34.000 --> 00:20:37.000 dy dx, r dr d theta, 00:20:37.000 --> 00:20:41.000 and then you will set up the bound carefully by slicing it 00:20:41.000 --> 00:20:45.000 and studying how the bounds for the inner variable depend on the 00:20:45.000 --> 00:20:51.000 outer variable. So, the first topic will be 00:20:51.000 --> 00:20:57.000 setting up double integrals. And so, remember, 00:20:57.000 --> 00:21:03.000 OK, so maybe I should make this more explicit. 00:21:03.000 --> 00:21:12.000 We want to draw a picture of R and take slices in the chosen 00:21:12.000 --> 00:21:18.000 way so that we get an iterated integral. 00:21:18.000 --> 00:21:25.000 OK, so let's do just a quick example. 00:21:25.000 --> 00:21:38.000 So, if I look at problem one on the exam 3B, 00:21:38.000 --> 00:21:43.000 it says to look at the line integral from zero to one, 00:21:43.000 --> 00:21:46.000 line integral from x to 2x of possibly something, 00:21:46.000 --> 00:21:50.000 but dy dx. And it says, 00:21:50.000 --> 00:21:58.000 let's look at how we would set this up the other way around by 00:21:58.000 --> 00:22:03.000 exchanging x and y. So, we should get to something 00:22:03.000 --> 00:22:06.000 that will be the same integral dx dy. 00:22:06.000 --> 00:22:09.000 I mean, if you have a function of x and y, then it will be the 00:22:09.000 --> 00:22:11.000 same function. But, of course, 00:22:11.000 --> 00:22:14.000 the bounds change. So, how do we exchange the 00:22:14.000 --> 00:22:17.000 order of integration? Well, the only way to do it 00:22:17.000 --> 00:22:20.000 consistently is to draw a picture. 00:22:20.000 --> 00:22:23.000 So, let's see, what does this mean? 00:22:23.000 --> 00:22:28.000 Here, it means we integrate from y equals x to y equals 2x, 00:22:28.000 --> 00:22:32.000 x between zero and one. So, we should draw a picture. 00:22:32.000 --> 00:22:35.000 The lower bound for y is y equals x. 00:22:35.000 --> 00:22:41.000 So, let's draw y equals x. That seems to be here. 00:22:41.000 --> 00:22:47.000 And, we'll go up to y equals 2x, which is a line also but 00:22:47.000 --> 00:22:52.000 with bigger slope. And then, all right, 00:22:52.000 --> 00:22:58.000 so for each value of x, my origin will go from x to 2x. 00:22:58.000 --> 00:23:03.000 Well, and I do this for all values of x that go to x equals 00:23:03.000 --> 00:23:06.000 one. So, I stop at x equals one, 00:23:06.000 --> 00:23:10.000 which is here. And then, my region is 00:23:10.000 --> 00:23:15.000 something like this. OK, so this point here, 00:23:15.000 --> 00:23:21.000 in case you are wondering, well, when x equals one, 00:23:21.000 --> 00:23:27.000 y is one. And that point here is one, two. 00:23:27.000 --> 00:23:29.000 OK, any questions about that so far? 00:23:29.000 --> 00:23:33.000 OK, so somehow that's the first kill, when you see an integral, 00:23:33.000 --> 00:23:36.000 how to figure out what it means, how to draw the region. 00:23:36.000 --> 00:23:39.000 And then there's a converse scale which is given the region, 00:23:39.000 --> 00:23:42.000 how to set up the integral for it. 00:23:42.000 --> 00:23:46.000 So, if we want to set up instead dx dy, 00:23:46.000 --> 00:23:50.000 then it means we are going to actually look at the converse 00:23:50.000 --> 00:23:54.000 question which is, for a given value of y, 00:23:54.000 --> 00:23:57.000 what is the range of values of x? 00:23:57.000 --> 00:24:01.000 OK, so if we fix y, well, where do we enter the 00:24:01.000 --> 00:24:04.000 region, and where do we leave it? 00:24:04.000 --> 00:24:08.000 So, we seem to enter on this side, and we seem to leave on 00:24:08.000 --> 00:24:10.000 that side. At least that seems to be true 00:24:10.000 --> 00:24:12.000 for the first few values of y that I choose. 00:24:12.000 --> 00:24:16.000 But, hey, if I take a larger value of y, then I will enter on 00:24:16.000 --> 00:24:19.000 the side, and I will leave on this vertical side, 00:24:19.000 --> 00:24:22.000 not on that one. So, I seem to have two 00:24:22.000 --> 00:24:28.000 different things going on. OK, the place where enter my 00:24:28.000 --> 00:24:38.000 region is always y equals 2x, which is the same as x equals y 00:24:38.000 --> 00:24:45.000 over two. So, x seems to always start at 00:24:45.000 --> 00:24:51.000 y over two. But, where I leave to be either 00:24:51.000 --> 00:24:55.000 x equals y, or here, x equals y. 00:24:55.000 --> 00:24:57.000 And, that depends on the value of y. 00:24:57.000 --> 00:24:59.000 So, in fact, I have to break this into two 00:24:59.000 --> 00:25:03.000 different integrals. I have to treat separately the 00:25:03.000 --> 00:25:07.000 case where y is between zero and one, and between one and two. 00:25:07.000 --> 00:25:15.000 So, what I do in that case is I just make two integrals. 00:25:15.000 --> 00:25:18.000 So, I say, both of them start at y over two. 00:25:18.000 --> 00:25:22.000 But, in the first case, we'll stop at x equals y. 00:25:22.000 --> 00:25:30.000 In the second case, we'll stop at x equals one. 00:25:30.000 --> 00:25:31.000 OK, and now, what are the values of y for 00:25:31.000 --> 00:25:34.000 each case? Well, the first case is when y 00:25:34.000 --> 00:25:38.000 is between zero and one. The second case is when y is 00:25:38.000 --> 00:25:40.000 between one and two, which I guess this picture now 00:25:40.000 --> 00:25:44.000 is completely unreadable, but hopefully you've been 00:25:44.000 --> 00:25:48.000 following what's going on, or else you can see it in the 00:25:48.000 --> 00:25:53.000 solutions to the problem. And, so that's our final answer. 00:25:53.000 --> 00:26:01.000 OK, any questions about how to set up double integrals in xy 00:26:01.000 --> 00:26:04.000 coordinates? No? 00:26:04.000 --> 00:26:07.000 OK, who feels comfortable with this kind of problem? 00:26:07.000 --> 00:26:11.000 OK, good. I'm happy to see the vast 00:26:11.000 --> 00:26:16.000 majority. So, the bad news is we have to 00:26:16.000 --> 00:26:23.000 be able to do it not only in xy coordinates, but also in polar 00:26:23.000 --> 00:26:27.000 coordinates. So, when you go to polar 00:26:27.000 --> 00:26:32.000 coordinates, basically all you have to remember on the side of 00:26:32.000 --> 00:26:36.000 integrand is that x becomes r cosine theta. 00:26:36.000 --> 00:26:45.000 Y becomes r sine theta. And, dx dy becomes r dr d theta. 00:26:45.000 --> 00:26:49.000 In terms of how you slice for your region, well, 00:26:49.000 --> 00:26:52.000 you will be integrating first over r. 00:26:52.000 --> 00:26:57.000 So, that means what you're doing is you're fixing the value 00:26:57.000 --> 00:26:59.000 of theta. And, for that value of theta, 00:26:59.000 --> 00:27:03.000 you ask yourself, for what range of values of r 00:27:03.000 --> 00:27:06.000 am I going to be inside my origin? 00:27:06.000 --> 00:27:09.000 So, if my origin looks like this, then for this value of 00:27:09.000 --> 00:27:13.000 theta, r would go from zero to whatever this distance is. 00:27:13.000 --> 00:27:16.000 And of course I have to find how this distance depends on 00:27:16.000 --> 00:27:18.000 theta. And then, I will find the 00:27:18.000 --> 00:27:20.000 extreme values of theta. Now, of course, 00:27:20.000 --> 00:27:22.000 is the origin is really looking like this, then you're not going 00:27:22.000 --> 00:27:25.000 to do it in polar coordinates. But, if it's like a circle or a 00:27:25.000 --> 00:27:27.000 half circle, or things like that, 00:27:27.000 --> 00:27:31.000 then even if a problem doesn't tell you to do it in polar 00:27:31.000 --> 00:27:34.000 coordinates you might want to seriously consider it. 00:27:34.000 --> 00:27:38.000 OK, so I'm not going to do it but problem two in the practice 00:27:38.000 --> 00:27:43.000 exam is a good example of doing something in polar coordinates. 00:27:43.000 --> 00:27:50.000 OK, so in terms of things that we 00:27:50.000 --> 00:27:56.000 do with double integrals, there's a few formulas that I'd 00:27:56.000 --> 00:28:00.000 like you to remember about applications that we've seen of 00:28:00.000 --> 00:28:04.000 double integrals. So, quantities that we can 00:28:04.000 --> 00:28:10.000 compute with double integrals include things like the area of 00:28:10.000 --> 00:28:13.000 region, its mass if it has a density, 00:28:13.000 --> 00:28:16.000 the average value of some function, 00:28:16.000 --> 00:28:19.000 for example, the average value of the x and 00:28:19.000 --> 00:28:22.000 y coordinates, which we called the center of 00:28:22.000 --> 00:28:31.000 mass or moments of inertia. So, these are just formulas to 00:28:31.000 --> 00:28:35.000 remember. So, for example, 00:28:35.000 --> 00:28:40.000 the area of region is the double integral of just dA, 00:28:40.000 --> 00:28:44.000 or if it helps you, one dA if you want. 00:28:44.000 --> 00:28:47.000 You are integrating the function 1. 00:28:47.000 --> 00:28:49.000 You have to remember formulas for mass, 00:28:49.000 --> 00:28:54.000 for the average value of a function is the F bar, 00:28:54.000 --> 00:29:05.000 in particular x bar y bar, which is the center of mass, 00:29:05.000 --> 00:29:14.000 and the moment of inertia. OK, so the polar moment of 00:29:14.000 --> 00:29:18.000 inertia, which is moment of inertia about the origin. 00:29:18.000 --> 00:29:22.000 OK, so that's double integral of x squared plus y squared, 00:29:22.000 --> 00:29:27.000 density dA, but also moments of inertia 00:29:27.000 --> 00:29:33.000 about the x and y axis, which are given by just taking 00:29:33.000 --> 00:29:36.000 one of these guys. Don't worry about moments of 00:29:36.000 --> 00:29:39.000 inertia about an arbitrary line. I will ask you for a moment of 00:29:39.000 --> 00:29:42.000 inertia for some weird line or something like that. 00:29:42.000 --> 00:29:47.000 OK, but these you should know. Now, what if you somehow, 00:29:47.000 --> 00:29:49.000 on the spur of the moment, you forget, what's the formula 00:29:49.000 --> 00:29:51.000 for moment of inertia? Well, I mean, 00:29:51.000 --> 00:29:54.000 I prefer if you know, but if you have a complete 00:29:54.000 --> 00:29:56.000 blank in your memory, there will still be partial 00:29:56.000 --> 00:29:59.000 credit were setting up the bounds and everything else. 00:29:59.000 --> 00:30:01.000 So, the general rule for the exam 00:30:01.000 --> 00:30:04.000 will be if you're stuck in a calculation or you're missing a 00:30:04.000 --> 00:30:08.000 little piece of the puzzle, try to do as much as you can. 00:30:08.000 --> 00:30:10.000 In particular, try to at least set up the 00:30:10.000 --> 00:30:15.000 bounds of the integral. There will be partial credit 00:30:15.000 --> 00:30:21.000 for that always. So, while we're at it about 00:30:21.000 --> 00:30:26.000 grand rules, how about evaluation? 00:30:26.000 --> 00:30:31.000 How about evaluating integrals? So, once you've set it up, 00:30:31.000 --> 00:30:33.000 you have to sometimes compute it. 00:30:33.000 --> 00:30:36.000 First of all, check just in case the problem 00:30:36.000 --> 00:30:40.000 says set up but do not evaluate. Then, don't waste your time 00:30:40.000 --> 00:30:45.000 evaluating it. If a problem says to compute 00:30:45.000 --> 00:30:50.000 it, then you have to compute it. So, what kinds of integration 00:30:50.000 --> 00:30:54.000 techniques do you need to know? So, you need to know, 00:30:54.000 --> 00:30:57.000 you must know, well, how to integrate the 00:30:57.000 --> 00:31:01.000 usual functions like one over x or x to the n, 00:31:01.000 --> 00:31:05.000 or exponential, sine, cosine, 00:31:05.000 --> 00:31:08.000 things like that, OK, so the usual integrals. 00:31:08.000 --> 00:31:16.000 You must know what I will call easy trigonometry. 00:31:16.000 --> 00:31:17.000 OK, I don't want to give you a complete list. 00:31:17.000 --> 00:31:20.000 And the more you ask me about which ones are on the list, 00:31:20.000 --> 00:31:22.000 the more I will add to the list. 00:31:22.000 --> 00:31:26.000 But, those that you know that you should know, 00:31:26.000 --> 00:31:28.000 you should know. Those that you think you 00:31:28.000 --> 00:31:31.000 shouldn't know, you don't have to know because 00:31:31.000 --> 00:31:36.000 I will say what I will say soon. You should know also 00:31:36.000 --> 00:31:41.000 substitution, how to set U equals something, 00:31:41.000 --> 00:31:45.000 and then see, oh, this becomes u times du, 00:31:45.000 --> 00:31:50.000 and so substitution method. What do I mean by easy 00:31:50.000 --> 00:31:52.000 trigonometrics? Well, certainly you should know 00:31:52.000 --> 00:31:54.000 how to ingrate sine. You should know how to 00:31:54.000 --> 00:31:57.000 integrate cosine. You should be aware that sine 00:31:57.000 --> 00:32:01.000 squared plus cosine squared simplifies to one. 00:32:01.000 --> 00:32:03.000 And, you should be aware of general things like that. 00:32:03.000 --> 00:32:06.000 I would like you to know, maybe, the double angles, 00:32:06.000 --> 00:32:09.000 sine 2x and cosine 2x. Know what these are, 00:32:09.000 --> 00:32:12.000 and the kinds of the easy things you can do with that, 00:32:12.000 --> 00:32:16.000 also things that involve substitution setting like U 00:32:16.000 --> 00:32:19.000 equals sine T or U equals cosine T. 00:32:19.000 --> 00:32:21.000 I mean, let me, instead, give an example of 00:32:21.000 --> 00:32:25.000 hard trig that you don't need to know, and then I will answer. 00:32:25.000 --> 00:32:34.000 OK, so, not needed on Thursday; it doesn't mean that I don't 00:32:34.000 --> 00:32:37.000 want you to know them. I would love you to know every 00:32:37.000 --> 00:32:41.000 single integral formula. But, that shouldn't be your top 00:32:41.000 --> 00:32:44.000 priority. So, you don't need to know 00:32:44.000 --> 00:32:47.000 things like hard trigonometric ones. 00:32:47.000 --> 00:32:52.000 So, let me give you an example. OK, so if I ask you to do this 00:32:52.000 --> 00:32:55.000 one, then actually I will give you maybe, you know, 00:32:55.000 --> 00:32:59.000 I will reprint the formula from the notes or something like 00:32:59.000 --> 00:33:02.000 that. OK, so that one you don't need 00:33:02.000 --> 00:33:04.000 to know. I would love if you happen to 00:33:04.000 --> 00:33:07.000 know it, but if you need it, it will be given to you. 00:33:07.000 --> 00:33:13.000 So, these kinds of things that you cannot compute by any easy 00:33:13.000 --> 00:33:16.000 method. And, integration by parts, 00:33:16.000 --> 00:33:21.000 I believe that I successfully test-solved all the problems 00:33:21.000 --> 00:33:26.000 without doing any single integration by parts. 00:33:26.000 --> 00:33:29.000 Again, in general, it's something that I would 00:33:29.000 --> 00:33:33.000 like you to know, but it shouldn't be a top 00:33:33.000 --> 00:33:40.000 priority for this week. OK, sorry, you had a question, 00:33:40.000 --> 00:33:42.000 or? Inverse trigonometric 00:33:42.000 --> 00:33:45.000 functions: let's say the most easy ones. 00:33:45.000 --> 00:33:50.000 I would like you to know the easiest inverse trig functions, 00:33:50.000 --> 00:33:56.000 but not much. OK, OK, so be aware that these 00:33:56.000 --> 00:34:04.000 functions exist, but it's not a top priority. 00:34:04.000 --> 00:34:06.000 I should say, the more I tell you I don't 00:34:06.000 --> 00:34:08.000 need you to know, the more your physics and other 00:34:08.000 --> 00:34:11.000 teachers might complain that, oh, these guys don't know how 00:34:11.000 --> 00:34:12.000 to integrate. So, try not to forget 00:34:12.000 --> 00:34:19.000 everything. But, yes? 00:34:19.000 --> 00:34:22.000 No, no, here I just mean for evaluating just a single 00:34:22.000 --> 00:34:24.000 variable integral. I will get to change variables 00:34:24.000 --> 00:34:27.000 and Jacobian soon, but I'm thinking of this as a 00:34:27.000 --> 00:34:29.000 different topic. What I mean by this one is if 00:34:29.000 --> 00:34:32.000 I'm asking you to integrate, I don't know, 00:34:32.000 --> 00:34:37.000 what's a good example? Zero to one t dt over square 00:34:37.000 --> 00:34:42.000 root of one plus t squared, then you should think of maybe 00:34:42.000 --> 00:34:44.000 substituting u equals one plus t squared, 00:34:44.000 --> 00:34:55.000 and then it becomes easier. OK, so this kind of trig, 00:34:55.000 --> 00:35:00.000 that's what I have in mind here specifically. 00:35:00.000 --> 00:35:02.000 And again, if you're stuck, 00:35:02.000 --> 00:35:05.000 in particular, if you hit this dreaded guy, 00:35:05.000 --> 00:35:09.000 and you don't actually have a formula giving you what it is, 00:35:09.000 --> 00:35:12.000 it means one of two things. One is something's wrong with 00:35:12.000 --> 00:35:13.000 your solution. The other option is something 00:35:13.000 --> 00:35:16.000 is wrong with my problem. So, either way, 00:35:16.000 --> 00:35:22.000 check quickly what you've done it if you can't find a mistake, 00:35:22.000 --> 00:35:27.000 then just move ahead to the next problem. 00:35:27.000 --> 00:35:30.000 Which one, this one? Yeah, 00:35:30.000 --> 00:35:32.000 I mean if you can do it, if you know how to do it, 00:35:32.000 --> 00:35:33.000 which everything is fair: I mean, 00:35:33.000 --> 00:35:36.000 generally speaking, give enough of it so that you 00:35:36.000 --> 00:35:38.000 found the solution by yourself, not like, 00:35:38.000 --> 00:35:43.000 you know, it didn't somehow come to you by magic. 00:35:43.000 --> 00:35:47.000 But, yeah, if you know how to integrate this without doing the 00:35:47.000 --> 00:35:49.000 substitution, that's absolutely fine by me. 00:35:49.000 --> 00:35:53.000 Just show enough work. The general rule is show enough 00:35:53.000 --> 00:35:58.000 work that we see that you knew what you are doing. 00:35:58.000 --> 00:36:02.000 OK, now another thing we've seen with double integrals is 00:36:02.000 --> 00:36:05.000 how to do more complicated changes of variables. 00:36:18.000 --> 00:36:23.000 So, when you want to replace x and y by some variables, 00:36:23.000 --> 00:36:28.000 u and v, given by some formulas in terms of x and y. 00:36:28.000 --> 00:36:33.000 So, you need to remember basically how to do them. 00:36:33.000 --> 00:36:36.000 So, you need to remember that the method consists of three 00:36:36.000 --> 00:36:43.000 steps. So, one is you have to find the 00:36:43.000 --> 00:36:46.000 Jacobian. And, you can choose to do 00:36:46.000 --> 00:36:50.000 either this Jacobian or the inverse one depending on what's 00:36:50.000 --> 00:36:53.000 easiest given what you're given. You don't have to worry about 00:36:53.000 --> 00:36:55.000 solving for things the other way around. 00:36:55.000 --> 00:36:58.000 Just compute one of these Jacobians. 00:36:58.000 --> 00:37:06.000 And then, the rule is that du dv is absolute value of the 00:37:06.000 --> 00:37:12.000 Jacobian dx dy. So, that takes care of dx dy, 00:37:12.000 --> 00:37:18.000 how to convert that into du dv. The second thing to know is 00:37:18.000 --> 00:37:20.000 that, well, 00:37:20.000 --> 00:37:25.000 you need to of course substitute any x and y's in the 00:37:25.000 --> 00:37:32.000 integrand to convert them to u's and v's so that you have a valid 00:37:32.000 --> 00:37:36.000 integrand involving only u and v. 00:37:36.000 --> 00:37:51.000 And then, the last part is setting up the bounds. 00:37:51.000 --> 00:37:54.000 And you see that, probably you seen on P-sets and 00:37:54.000 --> 00:37:58.000 an example we did in the lecture that this can be complicated. 00:37:58.000 --> 00:38:00.000 But now, in real life, you do this actually to 00:38:00.000 --> 00:38:02.000 simplify the integrals. So, 00:38:02.000 --> 00:38:04.000 probably the one that will be there on Thursday, 00:38:04.000 --> 00:38:07.000 if there's a problem about that on Thursday, 00:38:07.000 --> 00:38:10.000 it will be a situation where the bounds that you get after 00:38:10.000 --> 00:38:13.000 changing variables are reasonably easy. 00:38:13.000 --> 00:38:15.000 OK, I'm not saying that it will be completely obvious 00:38:15.000 --> 00:38:17.000 necessarily, but it will be a fairly easy situation. 00:38:17.000 --> 00:38:22.000 So, the general method is you look at your region, 00:38:22.000 --> 00:38:25.000 R, and it might have various sides. 00:38:25.000 --> 00:38:29.000 Well, on each side you ask yourself, what do I know about x 00:38:29.000 --> 00:38:33.000 and y, and how to convert that in terms of u and v? 00:38:33.000 --> 00:38:37.000 And maybe you'll find that the equation might be just u equals 00:38:37.000 --> 00:38:39.000 zero for example, or u equals v, 00:38:39.000 --> 00:38:42.000 or something like that. And then, it's up to you to 00:38:42.000 --> 00:38:46.000 decide what you want to do. But, maybe the easiest usually 00:38:46.000 --> 00:38:49.000 is to draw a new picture in terms of u and v coordinates of 00:38:49.000 --> 00:38:53.000 what your region will look like in the new coordinates. 00:38:53.000 --> 00:38:55.000 It might be that it will actually much easier. 00:38:55.000 --> 00:39:00.000 It should be easier looking than what you started with. 00:39:00.000 --> 00:39:05.000 OK, so that's the general idea. There is one change of variable 00:39:05.000 --> 00:39:09.000 problem on each of the two practice exams to give you a 00:39:09.000 --> 00:39:13.000 feeling for what's realistic. The problem that's on practice 00:39:13.000 --> 00:39:18.000 exam 3B actually is on the hard side of things because the 00:39:18.000 --> 00:39:21.000 question is kind of hidden in a way. 00:39:21.000 --> 00:39:25.000 So, if you look at problem six, you might find that it's not 00:39:25.000 --> 00:39:28.000 telling you very clearly what you have to do. 00:39:28.000 --> 00:39:34.000 That's because it was meant to be the hardest problem on that 00:39:34.000 --> 00:39:37.000 test. But, once you've reduced it to 00:39:37.000 --> 00:39:41.000 an actual change of variables problem, I expect you to be able 00:39:41.000 --> 00:39:44.000 to know how to do it. And, on practice exam 3A, 00:39:44.000 --> 00:39:48.000 there's also, I think it's problem five on 00:39:48.000 --> 00:39:52.000 the other practice exam. And, that one is actually 00:39:52.000 --> 00:39:55.000 pretty standard and straightforward. 00:39:55.000 --> 00:40:00.000 OK, time to move on, sorry. So, we've also seen about line 00:40:00.000 --> 00:40:00.000 integrals. 00:40:21.000 --> 00:40:30.000 OK, so line integrals, 00:40:30.000 --> 00:40:33.000 so the main thing to know about them, 00:40:33.000 --> 00:40:37.000 so the line integral for work, which is line integral of F.dr, 00:40:37.000 --> 00:40:40.000 so let's say that your vector field has components, 00:40:40.000 --> 00:40:49.000 M and N. So, the line integral for work 00:40:49.000 --> 00:40:57.000 becomes in coordinates integral of Mdx plus Ndy while we've also 00:40:57.000 --> 00:41:05.000 seen line integral for flux. So, line integral of F.n ds 00:41:05.000 --> 00:41:13.000 becomes the integral along C just to make sure that I give it 00:41:13.000 --> 00:41:18.000 to you correctly. So, remember that just, 00:41:18.000 --> 00:41:22.000 I don't want to make the mistake in front of you. 00:41:22.000 --> 00:41:30.000 So, T ds is dx, dy. And, the normal vector, 00:41:30.000 --> 00:41:36.000 so, T ds goes along the curve. Nds goes clockwise 00:41:36.000 --> 00:41:41.000 perpendicular to the curve. So, it's going to be, 00:41:41.000 --> 00:41:48.000 well, it's going to be dy and negative dx. 00:41:48.000 --> 00:42:00.000 So, you will be integrating negative Ndx plus Mdy. 00:42:00.000 --> 00:42:04.000 OK, see, if you are blanking and don't remember the signs, 00:42:04.000 --> 00:42:07.000 then you can just draw this picture and make sure that you 00:42:07.000 --> 00:42:10.000 get it right. So, you should know a little 00:42:10.000 --> 00:42:14.000 bit about geometric interpretation and how to see 00:42:14.000 --> 00:42:17.000 easily that it's going to be zero in some cases. 00:42:17.000 --> 00:42:21.000 But, mostly you should know how to compute, set up and compute 00:42:21.000 --> 00:42:23.000 these things. So, what do we do when we are 00:42:23.000 --> 00:42:24.000 here? Well, it's year, 00:42:24.000 --> 00:42:27.000 we have both x and y together, but we want to, 00:42:27.000 --> 00:42:30.000 because it's the line integral, there should be only one 00:42:30.000 --> 00:42:34.000 variable. So, the important thing to know 00:42:34.000 --> 00:42:39.000 is we want to reduce everything to a single parameter. 00:42:39.000 --> 00:42:55.000 OK, so the evaluation method is always by reducing to a single 00:42:55.000 --> 00:43:01.000 parameter. So, for example, 00:43:01.000 --> 00:43:06.000 maybe x and y are both functions of some variable, 00:43:06.000 --> 00:43:10.000 t, and then express everything in 00:43:10.000 --> 00:43:18.000 terms of some integral of, some quantity involving t dt. 00:43:18.000 --> 00:43:21.000 It could be that you will just express everything in terms of x 00:43:21.000 --> 00:43:24.000 or in terms of y, or in terms of some angle or 00:43:24.000 --> 00:43:26.000 something. It's up to you to choose how to 00:43:26.000 --> 00:43:29.000 parameterize things. And then, when you're there, 00:43:29.000 --> 00:43:33.000 it's a usual one variable integral with a single variable 00:43:33.000 --> 00:43:36.000 in there. OK, so that's the general 00:43:36.000 --> 00:43:40.000 method of calculation, but we've seen a shortcut for 00:43:40.000 --> 00:43:45.000 work when we can show that the field is the gradient of 00:43:45.000 --> 00:43:48.000 potential. So, 00:43:48.000 --> 00:43:55.000 one thing to know is if the curl of F, 00:43:55.000 --> 00:44:01.000 which is an x minus My happens to be zero, 00:44:01.000 --> 00:44:03.000 well, and now I can say, 00:44:03.000 --> 00:44:06.000 and the domain is simply connected, 00:44:06.000 --> 00:44:11.000 or if the field is defined everywhere, 00:44:11.000 --> 00:44:19.000 then F is actually a gradient field. 00:44:19.000 --> 00:44:22.000 So, that means, just to make it more concrete, 00:44:22.000 --> 00:44:26.000 that means we can find a function little f called the 00:44:26.000 --> 00:44:30.000 potential such that its derivative respect to x is M, 00:44:30.000 --> 00:44:32.000 and its derivative with respect to Y is N. 00:44:32.000 --> 00:44:37.000 We can solve these two conditions for the same 00:44:37.000 --> 00:44:42.000 function, f, simultaneously. And, how do we find this 00:44:42.000 --> 00:44:46.000 function, little f? OK, so that's the same as 00:44:46.000 --> 00:44:50.000 saying that the field, big F, is the gradient of 00:44:50.000 --> 00:44:52.000 little f. And, how do we find this 00:44:52.000 --> 00:44:54.000 function, little f? Well, we've seen two methods. 00:44:54.000 --> 00:44:58.000 One of them involves computing a line integral from the origin 00:44:58.000 --> 00:45:02.000 to a point in the plane by going first along the x axis, 00:45:02.000 --> 00:45:05.000 then vertically. The other method was to first 00:45:05.000 --> 00:45:09.000 figure out what this one tells us by integrating it with 00:45:09.000 --> 00:45:12.000 respect to x. And then, we differentiate our 00:45:12.000 --> 00:45:17.000 answer with respect to y, and we compare with that to get 00:45:17.000 --> 00:45:20.000 the complete answer. OK, so I is that relevant? 00:45:20.000 --> 00:45:22.000 Well, first of all it's relevant in 00:45:22.000 --> 00:45:25.000 physics, but it's also relevant just to 00:45:25.000 --> 00:45:29.000 calculation of line integrals because we see the fundamental 00:45:29.000 --> 00:45:34.000 theorem of calculus for line integrals which says if we are 00:45:34.000 --> 00:45:39.000 integrating a gradient field and we know what the potential is. 00:45:39.000 --> 00:45:43.000 Then, we just have to, well, the line integral is just 00:45:43.000 --> 00:45:46.000 the change in value of a potential. 00:45:46.000 --> 00:45:49.000 OK, so we take the value of a potential at the starting point, 00:45:49.000 --> 00:45:52.000 sorry, we take value potential at the endpoint minus the value 00:45:52.000 --> 00:45:58.000 at the starting point. And, that will give us the line 00:45:58.000 --> 00:46:00.000 integral, OK? So, important: 00:46:00.000 --> 00:46:05.000 this is only for work. There's no statement like that 00:46:05.000 --> 00:46:09.000 for flux, OK, so don't tried to fly this in a 00:46:09.000 --> 00:46:11.000 problem about flux. I mean, usually, 00:46:11.000 --> 00:46:13.000 if you look at the practice exams, 00:46:13.000 --> 00:46:17.000 you will see it's pretty clear that there's one problem in 00:46:17.000 --> 00:46:20.000 which you are supposed to do things this way. 00:46:20.000 --> 00:46:25.000 It's kind of a dead giveaway, but it's probably not too bad. 00:46:25.000 --> 00:46:29.000 OK, and the other thing we've seen, so I mentioned it at the 00:46:29.000 --> 00:46:32.000 beginning but let me mention it again. 00:46:32.000 --> 00:46:36.000 To compute things, Green's theorem, 00:46:36.000 --> 00:46:42.000 let's just compute, well, let us forget, 00:46:42.000 --> 00:46:45.000 sorry, find the value of a line integral along the closed curve 00:46:45.000 --> 00:46:47.000 by reducing it to double integral. 00:46:47.000 --> 00:46:55.000 So, the one for work says -- -- 00:46:55.000 --> 00:46:59.000 this, and you should remember that in 00:46:59.000 --> 00:47:01.000 there, so C is a closed curve that 00:47:01.000 --> 00:47:05.000 goes counterclockwise, and R is the region inside. 00:47:05.000 --> 00:47:08.000 So, the way you would, if you had to compute both 00:47:08.000 --> 00:47:10.000 sides separately, you would do them in extremely 00:47:10.000 --> 00:47:12.000 different ways, right? 00:47:12.000 --> 00:47:15.000 This one is a line integral. So, you use the method to 00:47:15.000 --> 00:47:18.000 explain here, namely, you express x and y in 00:47:18.000 --> 00:47:22.000 terms of a single variable. See that you're doing a circle. 00:47:22.000 --> 00:47:24.000 I want to see a theta. I don't want to see an R. 00:47:24.000 --> 00:47:27.000 R is not a variable. You are on the circle. 00:47:27.000 --> 00:47:30.000 This one is a double integral. So, if you are doing it, 00:47:30.000 --> 00:47:32.000 say, on a disk, you would have both R and theta 00:47:32.000 --> 00:47:34.000 if you're using polar coordinates. 00:47:34.000 --> 00:47:37.000 You would have both x and y. Here, you have two variables of 00:47:37.000 --> 00:47:40.000 integration. Here, you should have only one 00:47:40.000 --> 00:47:42.000 after you parameterize the curve. 00:47:42.000 --> 00:47:46.000 And, the fact that it stays curl F, I mean, 00:47:46.000 --> 00:47:51.000 curl F is just Nx-My is just like any function of x and y. 00:47:51.000 --> 00:47:54.000 OK, the fact that we called it curl F doesn't change how you 00:47:54.000 --> 00:47:56.000 compute it. You have first to compute the 00:47:56.000 --> 00:47:58.000 curl of F. Say you find, 00:47:58.000 --> 00:48:00.000 I don't know, xy minus x squared, 00:48:00.000 --> 00:48:04.000 well, it becomes just the usual double integral of the usual 00:48:04.000 --> 00:48:09.000 function xy minus x squared. There's nothing special to it 00:48:09.000 --> 00:48:15.000 because it's a curl. And, the other one is the 00:48:15.000 --> 00:48:21.000 counterpart for flux. So, it says this, 00:48:21.000 --> 00:48:25.000 and remember this is mx plus ny. 00:48:25.000 --> 00:48:27.000 I mean, what's important about these statements is not only 00:48:27.000 --> 00:48:30.000 remembering, you know, if you just know this formula 00:48:30.000 --> 00:48:32.000 by heart, you are still in trouble 00:48:32.000 --> 00:48:35.000 because you need to know what actually the symbols in here 00:48:35.000 --> 00:48:37.000 mean. So, you should remember, 00:48:37.000 --> 00:48:40.000 what is this line integral, and what's the divergence of a 00:48:40.000 --> 00:48:47.000 field? So, just something to remember. 00:48:47.000 --> 00:48:51.000 And, so I guess I'll let you figure out practice problems 00:48:51.000 --> 00:48:54.000 because it's time, but I think that's basically 00:48:54.000 --> 00:48:59.000 the list of all we've seen. And, well, that should be it.