1 00:00:01,000 --> 00:00:03,000 2 00:00:01,000 --> 00:00:03,000 The following content is provided under a Creative 3 00:00:03,000 --> 00:00:05,000 Commons license. Your support will help MIT 4 00:00:05,000 --> 00:00:08,000 OpenCourseWare continue to offer high quality educational 5 00:00:08,000 --> 00:00:13,000 resources for free. To make a donation or to view 6 00:00:13,000 --> 00:00:18,000 additional materials from hundreds of MIT courses, 7 00:00:18,000 --> 00:00:23,000 visit MIT OpenCourseWare at ocw.mit.edu. 8 00:00:24,000 --> 00:00:34,000 We were looking at vector fields last time. 9 00:00:34,000 --> 00:00:45,000 Last time we saw that if a vector field happens to be a 10 00:00:45,000 --> 00:00:56,000 gradient field -- -- then the line integral can be computed 11 00:00:56,000 --> 00:01:08,000 actually by taking the change in value of the potential between 12 00:01:08,000 --> 00:01:19,000 the end point and the starting point of the curve. 13 00:01:19,000 --> 00:01:24,000 If we have a curve c, from a point p0 to a point p1 14 00:01:24,000 --> 00:01:29,000 then the line integral for work depends only on the end points 15 00:01:29,000 --> 00:01:32,000 and not on the actual path we chose. 16 00:01:32,000 --> 00:01:43,000 We say that the line integral is path independent. 17 00:01:43,000 --> 00:01:49,000 And we also said that the vector field is conservative 18 00:01:49,000 --> 00:01:55,000 because of conservation of energy which tells you if you 19 00:01:55,000 --> 00:02:02,000 start at a point and you come back to the same point then you 20 00:02:02,000 --> 00:02:07,000 haven't gotten any work out of that force. 21 00:02:07,000 --> 00:02:15,000 If we have a closed curve then the line integral for work is 22 00:02:15,000 --> 00:02:18,000 just zero. And, basically, 23 00:02:18,000 --> 00:02:23,000 we say that these properties are equivalent being a gradient 24 00:02:23,000 --> 00:02:28,000 field or being path independent or being conservative. 25 00:02:28,000 --> 00:02:31,000 And what I promised to you is that today we would see a 26 00:02:31,000 --> 00:02:35,000 criterion to decide whether a vector field is a gradient field 27 00:02:35,000 --> 00:02:38,000 or not and how to find the potential function if it is a 28 00:02:38,000 --> 00:02:47,000 gradient field. So, that is the topic for today. 29 00:02:47,000 --> 00:03:00,000 The question is testing whether a given vector field, 30 00:03:00,000 --> 00:03:14,000 let's say M and N compliments, is a gradient field. 31 00:03:14,000 --> 00:03:16,000 For that, well, let's start with an 32 00:03:16,000 --> 00:03:26,000 observation. Say that it is a gradient field. 33 00:03:26,000 --> 00:03:31,000 That means that the first component of a field is just the 34 00:03:31,000 --> 00:03:35,000 partial of f with respect to some variable x and the second 35 00:03:35,000 --> 00:03:40,000 component is the partial of f with respect to y. 36 00:03:40,000 --> 00:03:43,000 Now we have seen an interesting property of the second partial 37 00:03:43,000 --> 00:03:46,000 derivatives of the function, which is if you take the 38 00:03:46,000 --> 00:03:49,000 partial derivative first with respect to x, 39 00:03:49,000 --> 00:03:52,000 then with respect to y, or first with respect to y, 40 00:03:52,000 --> 00:03:58,000 then with respect to x you get the same thing. 41 00:03:58,000 --> 00:04:07,000 We know f sub xy equals f sub yx, and that means M sub y 42 00:04:07,000 --> 00:04:12,000 equals N sub x. If you have a gradient field 43 00:04:12,000 --> 00:04:14,000 then it should have this property. 44 00:04:14,000 --> 00:04:17,000 You take the y component, take the derivative with 45 00:04:17,000 --> 00:04:19,000 respect to x, take the x component, 46 00:04:19,000 --> 00:04:20,000 differentiate with respect to y, 47 00:04:20,000 --> 00:04:31,000 you should get the same answer. And that is important to know. 48 00:04:31,000 --> 00:04:37,000 So, I am going to put that in a box. 49 00:04:37,000 --> 00:04:43,000 It is a broken box. The claim that I want to make 50 00:04:43,000 --> 00:04:45,000 is that there is a converse of sorts. 51 00:04:45,000 --> 00:04:47,000 This is actually basically all we need to check. 52 00:04:47,000 --> 00:05:06,000 53 00:05:06,000 --> 00:05:18,000 Conversely, if, and I am going to put here a 54 00:05:18,000 --> 00:05:33,000 condition, My equals Nx, then F is a gradient field. 55 00:05:33,000 --> 00:05:35,000 What is the condition that I need to put here? 56 00:05:35,000 --> 00:05:37,000 Well, we will see a more precise version of that next 57 00:05:37,000 --> 00:05:44,000 week. But for now let's just say if 58 00:05:44,000 --> 00:05:59,000 our vector field is defined and differentiable everywhere in the 59 00:05:59,000 --> 00:06:01,000 plane. We need, actually, 60 00:06:01,000 --> 00:06:04,000 a vector field that is well-defined everywhere. 61 00:06:04,000 --> 00:06:07,000 You are not allowed to have somehow places where it is not 62 00:06:07,000 --> 00:06:09,000 well-defined. Otherwise, actually, 63 00:06:09,000 --> 00:06:13,000 you have a counter example on your problem set this week. 64 00:06:13,000 --> 00:06:16,000 If you look at the last problem on the problem set this week, 65 00:06:16,000 --> 00:06:20,000 it gives you a vector field that satisfies this condition 66 00:06:20,000 --> 00:06:22,000 everywhere where it is defined. But, actually, 67 00:06:22,000 --> 00:06:24,000 there is a point where it is not defined. 68 00:06:24,000 --> 00:06:28,000 And that causes it, actually, to somehow -- I mean 69 00:06:28,000 --> 00:06:33,000 everything that I am going to say today breaks down for that 70 00:06:33,000 --> 00:06:36,000 example because of that. I mean, we will shed more light 71 00:06:36,000 --> 00:06:39,000 on this a bit later with the notion of simply connected 72 00:06:39,000 --> 00:06:42,000 regions and so on. But for now let's just say if 73 00:06:42,000 --> 00:06:47,000 it is defined everywhere and it satisfies this criterion then it 74 00:06:47,000 --> 00:06:52,000 is a gradient field. If you ignore the technical 75 00:06:52,000 --> 00:06:57,000 condition, being a gradient field means essentially the same 76 00:06:57,000 --> 00:07:11,000 thing as having this property. That is what we need to check. 77 00:07:11,000 --> 00:07:20,000 Let's look at an example. Well, one vector field that we 78 00:07:20,000 --> 00:07:24,000 have been looking at a lot was - yi xj. 79 00:07:24,000 --> 00:07:30,000 Remember that was the vector field that looked like a 80 00:07:30,000 --> 00:07:35,000 rotation at the unit speed. I think last time we already 81 00:07:35,000 --> 00:07:39,000 decided that this guy should not be allowed to be a gradient 82 00:07:39,000 --> 00:07:42,000 field and should not be conservative because if we 83 00:07:42,000 --> 00:07:45,000 integrate on the unit circle then we would get a positive 84 00:07:45,000 --> 00:07:49,000 answer. But let's check that indeed it 85 00:07:49,000 --> 00:07:55,000 fails our test. Well, let's call this M and 86 00:07:55,000 --> 00:08:01,000 let's call this guy N. If you look at partial M, 87 00:08:01,000 --> 00:08:07,000 partial y, that is going to be a negative one. 88 00:08:07,000 --> 00:08:11,000 If you take partial N, partial x, that is going to be 89 00:08:11,000 --> 00:08:12,000 one. These are not the same. 90 00:08:12,000 --> 00:08:17,000 So, indeed, this is not a gradient field. 91 00:08:17,000 --> 00:08:32,000 92 00:08:32,000 --> 00:08:53,000 Any questions about that? Yes? 93 00:08:53,000 --> 00:08:58,000 Your question is if I have the property M sub y equals N sub x 94 00:08:58,000 --> 00:09:03,000 only in a certain part of a plane for some values of x and 95 00:09:03,000 --> 00:09:06,000 y, can I conclude these things? 96 00:09:06,000 --> 00:09:09,000 And it is a gradient field in that part of the plane and 97 00:09:09,000 --> 00:09:13,000 conservative and so on. The answer for now is, 98 00:09:13,000 --> 00:09:17,000 in general, no. And when we spend a bit more 99 00:09:17,000 --> 00:09:20,000 time on it, actually, maybe I should move that up. 100 00:09:20,000 --> 00:09:24,000 Maybe we will talk about it later this week instead of when 101 00:09:24,000 --> 00:09:28,000 I had planned. There is a notion what it means 102 00:09:28,000 --> 00:09:30,000 for a region to be without holes. 103 00:09:30,000 --> 00:09:34,000 Basically, if you have that kind of property in a region 104 00:09:34,000 --> 00:09:38,000 that doesn't have any holes inside it then things will work. 105 00:09:38,000 --> 00:09:42,000 The problem comes from a vector field satisfying this criterion 106 00:09:42,000 --> 00:09:44,000 in a region but it has a hole in it. 107 00:09:44,000 --> 00:09:47,000 Because what you don't know is whether your potential is 108 00:09:47,000 --> 00:09:51,000 actually well-defined and takes the same value when you move all 109 00:09:51,000 --> 00:09:53,000 around the hole. It might come back to take a 110 00:09:53,000 --> 00:09:56,000 different value. If you look carefully and think 111 00:09:56,000 --> 00:10:00,000 hard about the example in the problem sets that is exactly 112 00:10:00,000 --> 00:10:04,000 what happens there. Again, I will say more about 113 00:10:04,000 --> 00:10:08,000 that later. For now we basically need our 114 00:10:08,000 --> 00:10:11,000 function to be, I mean, 115 00:10:11,000 --> 00:10:14,000 I should still say if you have this property for a vector field 116 00:10:14,000 --> 00:10:16,000 that is not quite defined everywhere, 117 00:10:16,000 --> 00:10:17,000 you are more than welcome, you know, 118 00:10:17,000 --> 00:10:20,000 you should probably still try to look for a potential using 119 00:10:20,000 --> 00:10:23,000 methods that we will see. But something might go wrong 120 00:10:23,000 --> 00:10:30,000 later. You might end up with a 121 00:10:30,000 --> 00:10:39,000 potential that is not well-defined. 122 00:10:39,000 --> 00:10:53,000 Let's do another example. Let's say that I give you this 123 00:10:53,000 --> 00:11:03,000 vector field. And this a here is a number. 124 00:11:03,000 --> 00:11:08,000 The question is for which value of a is this going to be 125 00:11:08,000 --> 00:11:13,000 possibly a gradient? If you have your flashcards 126 00:11:13,000 --> 00:11:17,000 then that is a good time to use them to vote, 127 00:11:17,000 --> 00:11:23,000 assuming that the number is small enough to be made with. 128 00:11:23,000 --> 00:11:27,000 Let's try to think about it. We want to call this guy M. 129 00:11:27,000 --> 00:11:35,000 We want to call that guy N. And we want to test M sub y 130 00:11:35,000 --> 00:11:42,000 versus N sub x. I don't see anyone. 131 00:11:42,000 --> 00:11:46,000 I see people doing it with their hands, and that works very 132 00:11:46,000 --> 00:11:48,000 well. OK. 133 00:11:48,000 --> 00:12:04,000 The question is for which value of a is this a gradient? 134 00:12:04,000 --> 00:12:10,000 I see various people with the correct answer. 135 00:12:10,000 --> 00:12:15,000 OK. That a strange answer. 136 00:12:15,000 --> 00:12:20,000 That is a good answer. OK. 137 00:12:20,000 --> 00:12:28,000 The vote seems to be for a equals eight. 138 00:12:28,000 --> 00:12:35,000 Let's see. What if I take M sub y? 139 00:12:35,000 --> 00:12:41,000 That is going to be just ax. And N sub x? 140 00:12:41,000 --> 00:12:47,000 That is 8x. I would like a equals eight. 141 00:12:47,000 --> 00:12:50,000 By the way, when you set these two equal to each other, 142 00:12:50,000 --> 00:12:52,000 they really have to be equal everywhere. 143 00:12:52,000 --> 00:12:55,000 You don't want to somehow solve for x or anything like that. 144 00:12:55,000 --> 00:12:59,000 You just want these expressions, in terms of x and 145 00:12:59,000 --> 00:13:02,000 y, to be the same quantities. I mean you cannot say if x 146 00:13:02,000 --> 00:13:07,000 equals z they are always equal. Yeah, that is true. 147 00:13:07,000 --> 00:13:13,000 But that is not what we are asking. 148 00:13:13,000 --> 00:13:18,000 Now we come to the next logical question. 149 00:13:18,000 --> 00:13:20,000 Let's say that we have passed the test. 150 00:13:20,000 --> 00:13:23,000 We have put a equals eight in here. 151 00:13:23,000 --> 00:13:26,000 Now it should be a gradient field. 152 00:13:26,000 --> 00:13:30,000 The question is how do we find the potential? 153 00:13:30,000 --> 00:13:36,000 That becomes eight from now on. The question is how do we find 154 00:13:36,000 --> 00:13:39,000 the function which has this as gradient? 155 00:13:39,000 --> 00:13:43,000 One option is to try to guess. Actually, quite often you will 156 00:13:43,000 --> 00:13:47,000 succeed that way. But that is not a valid method 157 00:13:47,000 --> 00:13:50,000 on next week's test. We are going to see two 158 00:13:50,000 --> 00:13:55,000 different systematic methods. And you should be using one of 159 00:13:55,000 --> 00:14:00,000 these because guessing doesn't always work. 160 00:14:00,000 --> 00:14:03,000 And, actually, I can come up with examples 161 00:14:03,000 --> 00:14:07,000 where if you try to guess you will surely fail. 162 00:14:07,000 --> 00:14:15,000 I can come up with trick ones, but I don't want to put that on 163 00:14:15,000 --> 00:14:24,000 the test. The next stage is finding the 164 00:14:24,000 --> 00:14:30,000 potential. And let me just emphasize that 165 00:14:30,000 --> 00:14:36,000 we can only do that if step one was successful. 166 00:14:36,000 --> 00:14:41,000 If we have a vector field that cannot possibly be a gradient 167 00:14:41,000 --> 00:14:45,000 then we shouldn't try to look for a potential. 168 00:14:45,000 --> 00:14:52,000 It is kind of obvious but is probably worth pointing out. 169 00:14:52,000 --> 00:15:00,000 There are two methods. The first method that we will 170 00:15:00,000 --> 00:15:16,000 see is computing line integrals. Let's see how that works. 171 00:15:16,000 --> 00:15:25,000 Let's say that I take some path that starts at the origin. 172 00:15:25,000 --> 00:15:26,000 Or, actually, anywhere you want, 173 00:15:26,000 --> 00:15:29,000 but let's take the origin. That is my favorite point. 174 00:15:29,000 --> 00:15:36,000 And let's go to a point with coordinates (x1, 175 00:15:36,000 --> 00:15:40,000 y1). And let's take my favorite 176 00:15:40,000 --> 00:15:45,000 curve and compute the line integral of that field, 177 00:15:45,000 --> 00:15:49,000 you know, the work done along the curve. 178 00:15:49,000 --> 00:15:55,000 Well, by the fundamental theorem, that should be equal to 179 00:15:55,000 --> 00:16:02,000 the value of the potential at the end point minus the value at 180 00:16:02,000 --> 00:16:09,000 the origin. That means I can actually write 181 00:16:09,000 --> 00:16:19,000 f of (x1, y1) equals -- -- that line integral plus the value at 182 00:16:19,000 --> 00:16:26,000 the origin. And that is just a constant. 183 00:16:26,000 --> 00:16:27,000 We don't know what it is. And, actually, 184 00:16:27,000 --> 00:16:30,000 we can choose what it is. Because if you have a 185 00:16:30,000 --> 00:16:33,000 potential, say that you have some potential function. 186 00:16:33,000 --> 00:16:34,000 And let's say that you add one to it. 187 00:16:34,000 --> 00:16:36,000 It is still a potential function. 188 00:16:36,000 --> 00:16:38,000 Adding one doesn't change the gradient. 189 00:16:38,000 --> 00:16:41,000 You can even add 18 or any number that you want. 190 00:16:41,000 --> 00:16:44,000 This is just going to be an integration constant. 191 00:16:44,000 --> 00:16:47,000 It is the same thing as, in one variable calculus, 192 00:16:47,000 --> 00:16:49,000 when you take the anti-derivative of a function it 193 00:16:49,000 --> 00:16:52,000 is only defined up to adding the constant. 194 00:16:52,000 --> 00:16:56,000 We have this integration constant, but apart from that we 195 00:16:56,000 --> 00:16:59,000 know that we should be able to get a potential from this. 196 00:16:59,000 --> 00:17:03,000 And this we can compute using the definition of the line 197 00:17:03,000 --> 00:17:06,000 integral. And we don't know what little f 198 00:17:06,000 --> 00:17:11,000 is, but we know what the vector field is so we can compute that. 199 00:17:11,000 --> 00:17:14,000 Of course, to do the calculation we probably don't 200 00:17:14,000 --> 00:17:18,000 want to use this kind of path. I mean if that is your favorite 201 00:17:18,000 --> 00:17:21,000 path then that is fine, but it is not very easy to 202 00:17:21,000 --> 00:17:24,000 compute the line integral along this, 203 00:17:24,000 --> 00:17:28,000 especially since I didn't tell you what the definition is. 204 00:17:28,000 --> 00:17:31,000 There are easier favorite paths to have. 205 00:17:31,000 --> 00:17:33,000 For example, you can go on a straight line 206 00:17:33,000 --> 00:17:37,000 from the origin to that point. That would be slightly easier. 207 00:17:37,000 --> 00:17:40,000 But then there is one easier. The easiest of all, 208 00:17:40,000 --> 00:17:47,000 probably, is to just go first along the x-axis to (x1,0) and 209 00:17:47,000 --> 00:17:51,000 then go up parallel to the y-axis. 210 00:17:51,000 --> 00:17:54,000 Why is that easy? Well, that is because when we 211 00:17:54,000 --> 00:17:57,000 do the line integral it becomes M dx N dy. 212 00:17:57,000 --> 00:18:05,000 And then, on each of these pieces, one-half just goes away 213 00:18:05,000 --> 00:18:11,000 because x, y is constant. Let's try to use that method in 214 00:18:11,000 --> 00:18:12,000 our example. 215 00:18:12,000 --> 00:18:45,000 216 00:18:45,000 --> 00:18:56,000 Let's say that I want to go along this path from the origin, 217 00:18:56,000 --> 00:19:06,000 first along the x-axis to (x1,0) and then vertically to 218 00:19:06,000 --> 00:19:14,000 (x1, y1). And so I want to compute for 219 00:19:14,000 --> 00:19:21,000 the line integral along that curve. 220 00:19:21,000 --> 00:19:24,000 Let's say I want to do it for this vector field. 221 00:19:24,000 --> 00:19:33,000 I want to find the potential for this vector field. 222 00:19:33,000 --> 00:19:37,000 Let me copy it because I will have to erase at some point. 223 00:19:37,000 --> 00:19:50,000 4x squared plus 8xy and 3y squared plus 4x squared. 224 00:19:50,000 --> 00:19:59,000 That will become the integral of 4x squared plus 8 xy times dx 225 00:19:59,000 --> 00:20:05,000 plus 3y squared plus 4x squared times dy. 226 00:20:05,000 --> 00:20:08,000 To evaluate on this broken line, I will, 227 00:20:08,000 --> 00:20:13,000 of course, evaluate separately on each of the two segments. 228 00:20:13,000 --> 00:20:20,000 I will start with this segment that I will call c1 and then I 229 00:20:20,000 --> 00:20:25,000 will do this one that I will call c2. 230 00:20:25,000 --> 00:20:30,000 On c1, how do I evaluate my integral? 231 00:20:30,000 --> 00:20:38,000 Well, if I am on c1 then x varies from zero to x1. 232 00:20:38,000 --> 00:20:40,000 Well, actually, I don't know if x1 is positive 233 00:20:40,000 --> 00:20:41,000 or not so I shouldn't write this. 234 00:20:41,000 --> 00:20:48,000 I really should say just x goes from zero to x1. 235 00:20:48,000 --> 00:20:54,000 And what about y? y is just 0. 236 00:20:54,000 --> 00:21:00,000 I will set y equal to zero and also dy equal to zero. 237 00:21:00,000 --> 00:21:08,000 I get that the line integral on c1 -- Well, a lot of stuff goes 238 00:21:08,000 --> 00:21:11,000 away. The entire second term with dy 239 00:21:11,000 --> 00:21:15,000 goes away because dy is zero. And, in the first term, 240 00:21:15,000 --> 00:21:18,000 8xy goes away because y is zero as well. 241 00:21:18,000 --> 00:21:27,000 I just have an integral of 4x squared dx from zero to x1. 242 00:21:27,000 --> 00:21:31,000 By the way, now you see why I have been using an x1 and a y1 243 00:21:31,000 --> 00:21:33,000 for my point and not just x and y. 244 00:21:33,000 --> 00:21:36,000 It is to avoid confusion. I am using x and y as my 245 00:21:36,000 --> 00:21:41,000 integration variables and x1, y1 as constants that are 246 00:21:41,000 --> 00:21:45,000 representing the end point of my path. 247 00:21:45,000 --> 00:21:51,000 And so, if I integrate this, I should get four-thirds x1 248 00:21:51,000 --> 00:21:54,000 cubed. That is the first part. 249 00:21:54,000 --> 00:22:01,000 Next I need to do the second segment. 250 00:22:01,000 --> 00:22:09,000 If I am on c2, y goes from zero to y1. 251 00:22:09,000 --> 00:22:16,000 And what about x? x is constant equal to x1 so dx 252 00:22:16,000 --> 00:22:22,000 becomes just zero. It is a constant. 253 00:22:22,000 --> 00:22:30,000 If I take the line integral of c2, F dot dr then I will get the 254 00:22:30,000 --> 00:22:37,000 integral from zero to y1. The entire first term with dx 255 00:22:37,000 --> 00:22:47,000 goes away and then I have 3y squared plus 4x1 squared times 256 00:22:47,000 --> 00:22:52,000 dy. That integrates to y cubed plus 257 00:22:52,000 --> 00:23:01,000 4x1 squared y from zero to y1. Or, if you prefer, 258 00:23:01,000 --> 00:23:11,000 that is y1 cubed plus 4x1 squared y1. 259 00:23:11,000 --> 00:23:15,000 Now that we have done both of them we can just add them 260 00:23:15,000 --> 00:23:19,000 together, and that will give us the formula for the potential. 261 00:23:19,000 --> 00:23:40,000 262 00:23:40,000 --> 00:23:50,000 F of x1 and y1 is four-thirds x1 cubed plus y1 cubed plus 4x1 263 00:23:50,000 --> 00:23:57,000 squared y1 plus a constant. That constant is just the 264 00:23:57,000 --> 00:24:03,000 integration constant that we had from the beginning. 265 00:24:03,000 --> 00:24:05,000 Now you can drop the subscripts if you prefer. 266 00:24:05,000 --> 00:24:14,000 You can just say f is four-thirds x cubed plus y cubed 267 00:24:14,000 --> 00:24:20,000 plus 4x squared y plus constant. And you can check. 268 00:24:20,000 --> 00:24:25,000 If you take the gradient of this, you should get again this 269 00:24:25,000 --> 00:24:29,000 vector field over there. Any questions about this method? 270 00:24:29,000 --> 00:24:33,000 Yes? No. 271 00:24:33,000 --> 00:24:35,000 Well, it depends whether you are just trying to find one 272 00:24:35,000 --> 00:24:38,000 potential or if you are trying to find all the possible 273 00:24:38,000 --> 00:24:40,000 potentials. If a problem just says find a 274 00:24:40,000 --> 00:24:43,000 potential then you don't have to use the constant. 275 00:24:43,000 --> 00:24:47,000 This guy without the constant is a valid potential. 276 00:24:47,000 --> 00:24:52,000 You just have others. If your neighbor comes to you 277 00:24:52,000 --> 00:24:58,000 and say your answer must be wrong because I got this plus 278 00:24:58,000 --> 00:25:01,000 18, well, both answers are correct. 279 00:25:01,000 --> 00:25:05,000 By the way. Instead of going first along 280 00:25:05,000 --> 00:25:08,000 the x-axis vertically, you could do it the other way 281 00:25:08,000 --> 00:25:11,000 around. Of course, start along the 282 00:25:11,000 --> 00:25:15,000 y-axis and then horizontally. That is the same level of 283 00:25:15,000 --> 00:25:19,000 difficulty. You just exchange roles of x 284 00:25:19,000 --> 00:25:21,000 and y. In some cases, 285 00:25:21,000 --> 00:25:26,000 it is actually even making more sense maybe to go radially, 286 00:25:26,000 --> 00:25:30,000 start out from the origin to your end point. 287 00:25:30,000 --> 00:25:37,000 But usually this setting is easier just because each of 288 00:25:37,000 --> 00:25:43,000 these two guys were very easy to compute. 289 00:25:43,000 --> 00:25:46,000 But somehow maybe if you suspect that polar coordinates 290 00:25:46,000 --> 00:25:49,000 will be involved somehow in the answer then maybe it makes sense 291 00:25:49,000 --> 00:26:01,000 to choose different paths. Maybe a straight line is better. 292 00:26:01,000 --> 00:26:13,000 Now we have another method to look at which is using 293 00:26:13,000 --> 00:26:19,000 anti-derivatives. The goal is the same, 294 00:26:19,000 --> 00:26:21,000 still to find the potential function. 295 00:26:21,000 --> 00:26:26,000 And you see that finding the potential is really the 296 00:26:26,000 --> 00:26:31,000 multivariable analog of finding the anti-derivative in the one 297 00:26:31,000 --> 00:26:34,000 variable. Here we did it basically by 298 00:26:34,000 --> 00:26:38,000 hand by computing the integral. The other thing you could try 299 00:26:38,000 --> 00:26:39,000 to say is, wait, I already know how to take 300 00:26:39,000 --> 00:26:42,000 anti-derivatives. Let's use that instead of 301 00:26:42,000 --> 00:26:45,000 computing integrals. And it works but you have to be 302 00:26:45,000 --> 00:26:51,000 careful about how you do it. Let's see how that works. 303 00:26:51,000 --> 00:26:53,000 Let's still do it with the same example. 304 00:26:53,000 --> 00:27:02,000 We want to solve the equations. We want a function such that f 305 00:27:02,000 --> 00:27:13,000 sub x is 4x squared plus 8xy and f sub y is 3y squared plus 4x 306 00:27:13,000 --> 00:27:16,000 squared. Let's just look at one of these 307 00:27:16,000 --> 00:27:20,000 at a time. If we look at this one, 308 00:27:20,000 --> 00:27:28,000 well, we know how to solve this because it is just telling us we 309 00:27:28,000 --> 00:27:33,000 have to integrate this with respect to x. 310 00:27:33,000 --> 00:27:38,000 Well, let's call them one and two because I will have to refer 311 00:27:38,000 --> 00:27:43,000 to them again. Let's start with equation one 312 00:27:43,000 --> 00:27:48,000 and lets integrate with respect to x. 313 00:27:48,000 --> 00:27:51,000 Well, it tells us that f should be, 314 00:27:51,000 --> 00:27:55,000 what do I get when I integrate this with respect to x, 315 00:27:55,000 --> 00:28:02,000 four-thirds x cubed plus, when I integrate 8xy, 316 00:28:02,000 --> 00:28:08,000 y is just a constant, so I will get 4x squared y. 317 00:28:08,000 --> 00:28:11,000 And that is not quite the end to it because there is an 318 00:28:11,000 --> 00:28:15,000 integration constant. And here, when I say there is 319 00:28:15,000 --> 00:28:18,000 an integration constant, it just means the extra term 320 00:28:18,000 --> 00:28:21,000 does not depend on x. That is what it means to be a 321 00:28:21,000 --> 00:28:25,000 constant in this setting. But maybe my constant still 322 00:28:25,000 --> 00:28:28,000 depends on y so it is not actually a true constant. 323 00:28:28,000 --> 00:28:30,000 A constant that depends on y is not really a constant. 324 00:28:30,000 --> 00:28:38,000 It is actually a function of y. The good news that we have is 325 00:28:38,000 --> 00:28:40,000 that this function normally depends on x. 326 00:28:40,000 --> 00:28:46,000 We have made some progress. We have part of the answer and 327 00:28:46,000 --> 00:28:53,000 we have simplified the problem. If we have anything that looks 328 00:28:53,000 --> 00:28:56,000 like this, it will satisfy the first condition. 329 00:28:56,000 --> 00:28:59,000 Now we need to look at the second condition. 330 00:28:59,000 --> 00:29:12,000 We want f sub y to be that. But we know what f is, 331 00:29:12,000 --> 00:29:15,000 so let's compute f sub y from this. 332 00:29:15,000 --> 00:29:20,000 From this I get f sub y. What do I get if I 333 00:29:20,000 --> 00:29:22,000 differentiate this with respect to y? 334 00:29:22,000 --> 00:29:37,000 Well, I get zero plus 4x squared plus the derivative of 335 00:29:37,000 --> 00:29:46,000 g. I would like to match this with 336 00:29:46,000 --> 00:29:51,000 what I had. If I match this with equation 337 00:29:51,000 --> 00:29:55,000 two then that will tell me what the derivative of g should be. 338 00:29:55,000 --> 00:30:15,000 339 00:30:15,000 --> 00:30:20,000 If we compare the two things there, we get 4x squared plus g 340 00:30:20,000 --> 00:30:26,000 prime of y should be equal to 3y squared by 4x squared. 341 00:30:26,000 --> 00:30:31,000 And, of course, the 4x squares go away. 342 00:30:31,000 --> 00:30:35,000 That tells you g prime is 3y squared. 343 00:30:35,000 --> 00:30:42,000 And that integrates to y cubed plus constant. 344 00:30:42,000 --> 00:30:46,000 Now, this time the constant is a true constant because g did 345 00:30:46,000 --> 00:30:48,000 not depend on anything other than y. 346 00:30:48,000 --> 00:30:54,000 And the constant does not depend on y so it is a real 347 00:30:54,000 --> 00:30:58,000 constant now. Now we just plug this back into 348 00:30:58,000 --> 00:31:05,000 this guy. Let's call him star. 349 00:31:05,000 --> 00:31:13,000 If we plug this into star, we get f equals four-thirds x 350 00:31:13,000 --> 00:31:21,000 cubed plus 4x squared y plus y cubed plus constant. 351 00:31:21,000 --> 00:31:30,000 I mean, of course, again, now this constant is 352 00:31:30,000 --> 00:31:33,000 optional. The advantage of this method is 353 00:31:33,000 --> 00:31:35,000 you don't have to write any integrals. 354 00:31:35,000 --> 00:31:40,000 The small drawback is you have to follow this procedure 355 00:31:40,000 --> 00:31:45,000 carefully. By the way, one common pitfall 356 00:31:45,000 --> 00:31:48,000 that is tempting. After you have done this, 357 00:31:48,000 --> 00:31:51,000 what is very tempting is to just say, well, 358 00:31:51,000 --> 00:31:53,000 let's do the same with this guy. 359 00:31:53,000 --> 00:31:55,000 Let's integrate this with respect to y. 360 00:31:55,000 --> 00:31:58,000 You will get another expression for f up to a constant that 361 00:31:58,000 --> 00:32:01,000 depends on x. And then let's match them. 362 00:32:01,000 --> 00:32:04,000 Well, the difficulty is matching is actually quite 363 00:32:04,000 --> 00:32:09,000 tricky because you don't know in advance whether they will be the 364 00:32:09,000 --> 00:32:13,000 same expression. It could be you could say let's 365 00:32:13,000 --> 00:32:16,000 just take the terms that are here and missing there and 366 00:32:16,000 --> 00:32:20,000 combine the terms, you know, take all the terms 367 00:32:20,000 --> 00:32:23,000 that appear in either one. That is actually not a good way 368 00:32:23,000 --> 00:32:25,000 to do it, because if I put sufficiently 369 00:32:25,000 --> 00:32:28,000 complicated trig functions in there then you might not be able 370 00:32:28,000 --> 00:32:30,000 to see that two terms are the same. 371 00:32:30,000 --> 00:32:34,000 Take an easy one. Let's say that here I have one 372 00:32:34,000 --> 00:32:40,000 plus tangent square and here I have a secan square then you 373 00:32:40,000 --> 00:32:46,000 might not actually notice that there is a difference. 374 00:32:46,000 --> 00:32:50,000 But there is no difference. Whatever. 375 00:32:50,000 --> 00:32:54,000 Anyway, I am saying do it this way, don't do it any other way 376 00:32:54,000 --> 00:32:57,000 because there is a risk of making a mistake otherwise. 377 00:32:57,000 --> 00:33:00,000 I mean, on the other hand, you could start with 378 00:33:00,000 --> 00:33:03,000 integrating with respect to y and then differentiate and match 379 00:33:03,000 --> 00:33:06,000 with respect to x. But what I am saying is just 380 00:33:06,000 --> 00:33:09,000 take one of them, integrate, 381 00:33:09,000 --> 00:33:12,000 get an answer that involves a function of the other variable, 382 00:33:12,000 --> 00:33:18,000 then differentiate that answer and compare and see what you 383 00:33:18,000 --> 00:33:21,000 get. By the way, here, 384 00:33:21,000 --> 00:33:27,000 of course, after we simplified there were only y's here. 385 00:33:27,000 --> 00:33:29,000 There were no x's. And that is kind of good news. 386 00:33:29,000 --> 00:33:33,000 I mean, if you had had an x here in this expression that 387 00:33:33,000 --> 00:33:36,000 would have told you that something is going wrong. 388 00:33:36,000 --> 00:33:39,000 g is a function of y only. If you get an x here, 389 00:33:39,000 --> 00:33:42,000 maybe you want to go back and check whether it is really a 390 00:33:42,000 --> 00:33:47,000 gradient field. Yes? 391 00:33:47,000 --> 00:33:49,000 Yes, this will work with functions of more than two 392 00:33:49,000 --> 00:33:51,000 variables. Both methods work with more 393 00:33:51,000 --> 00:33:53,000 than two variables. We are going to see it in the 394 00:33:53,000 --> 00:33:56,000 case where more than two means three. 395 00:33:56,000 --> 00:34:00,000 We are going to see that in two or three weeks from now. 396 00:34:00,000 --> 00:34:04,000 I mean, basically starting at the end of next week, 397 00:34:04,000 --> 00:34:08,000 we are going to do triple integrals, line integrals in 398 00:34:08,000 --> 00:34:10,000 space and so on. The format is first we do 399 00:34:10,000 --> 00:34:13,000 everything in two variables. Then we will do three variables. 400 00:34:13,000 --> 00:34:20,000 And then what happens with more than three will be left to your 401 00:34:20,000 --> 00:34:25,000 imagination. Any other questions about 402 00:34:25,000 --> 00:34:29,000 either of these methods? A quick poll. 403 00:34:29,000 --> 00:34:34,000 Who prefers the first method? Who prefers the second method? 404 00:34:34,000 --> 00:34:41,000 Wow. OK. 405 00:34:41,000 --> 00:34:45,000 Anyway, you will get to use whichever one you want. 406 00:34:45,000 --> 00:34:47,000 And I would agree with you, but the second method is 407 00:34:47,000 --> 00:34:50,000 slightly more effective in that you are writing less stuff. 408 00:34:50,000 --> 00:34:54,000 You don't have to set up all these line integrals. 409 00:34:54,000 --> 00:35:03,000 On the other hand, it does require a little bit 410 00:35:03,000 --> 00:35:19,000 more attention. Let's move on a bit. 411 00:35:19,000 --> 00:35:24,000 Let me start by actually doing a small recap. 412 00:35:24,000 --> 00:35:38,000 We said we have various notions. One is to say that the vector 413 00:35:38,000 --> 00:35:48,000 field is a gradient in a certain region of a plane. 414 00:35:48,000 --> 00:35:54,000 And we have another notion which is being conservative. 415 00:35:54,000 --> 00:36:06,000 It says that the line integral is zero along any closed curve. 416 00:36:06,000 --> 00:36:10,000 Actually, let me introduce a new piece of notation. 417 00:36:10,000 --> 00:36:14,000 To remind ourselves that we are doing it along a closed curve, 418 00:36:14,000 --> 00:36:18,000 very often we put just a circle for the integral to tell us this 419 00:36:18,000 --> 00:36:21,000 is a curve that closes on itself. 420 00:36:21,000 --> 00:36:25,000 It ends where it started. I mean it doesn't change 421 00:36:25,000 --> 00:36:28,000 anything concerning the definition or how you compute it 422 00:36:28,000 --> 00:36:31,000 or anything. It just reminds you that you 423 00:36:31,000 --> 00:36:34,000 are doing it on a closed curve. It is actually useful for 424 00:36:34,000 --> 00:36:37,000 various physical applications. And also, when you state 425 00:36:37,000 --> 00:36:41,000 theorems in that way, it reminds you,oh.. 426 00:36:41,000 --> 00:36:45,000 I need to be on a closed curve to do it. 427 00:36:45,000 --> 00:36:51,000 And so we have said these two things are equivalent. 428 00:36:51,000 --> 00:37:00,000 Now we have a third thing which is N sub x equals M sub y at 429 00:37:00,000 --> 00:37:03,000 every point. Just to summarize the 430 00:37:03,000 --> 00:37:06,000 discussion. We have said if we have a 431 00:37:06,000 --> 00:37:09,000 gradient field then we have this. 432 00:37:09,000 --> 00:37:18,000 And the converse is true in suitable regions. 433 00:37:18,000 --> 00:37:32,000 We have a converse if F is defined in the entire plane. 434 00:37:32,000 --> 00:37:43,000 Or, as we will see soon, in a simply connected region. 435 00:37:43,000 --> 00:37:45,000 I guess some of you cannot see what I am writing here, 436 00:37:45,000 --> 00:37:48,000 but it doesn't matter because you are not officially supposed 437 00:37:48,000 --> 00:37:53,000 to know it yet. That will be next week. 438 00:37:53,000 --> 00:37:57,000 Anyway, I said the fact that Nx equals 439 00:37:57,000 --> 00:38:01,000 My implies that we have a gradient field and is only if a 440 00:38:01,000 --> 00:38:06,000 vector field is defined in the entire plane or in a region that 441 00:38:06,000 --> 00:38:12,000 is called simply connected. And more about that later. 442 00:38:12,000 --> 00:38:17,000 Now let me just introduce a quantity that probably a lot of 443 00:38:17,000 --> 00:38:22,000 you have heard about in physics that measures precisely fairly 444 00:38:22,000 --> 00:38:26,000 ought to be conservative. That is called the curl of a 445 00:38:26,000 --> 00:38:27,000 vector field. 446 00:38:27,000 --> 00:39:06,000 447 00:39:06,000 --> 00:39:19,000 For the definition we say that the curl of F is the quantity N 448 00:39:19,000 --> 00:39:27,000 sub x - M sub y. It is just replicating the 449 00:39:27,000 --> 00:39:35,000 information we had but in a way that is a single quantity. 450 00:39:35,000 --> 00:39:43,000 In this new language, the conditions that we had over 451 00:39:43,000 --> 00:39:50,000 there, this condition says curl F equals zero. 452 00:39:50,000 --> 00:39:56,000 That is the new version of Nx equals My. 453 00:39:56,000 --> 00:40:06,000 It measures failure of a vector field to be conservative. 454 00:40:06,000 --> 00:40:21,000 The test for conservativeness is that the curl of F should be 455 00:40:21,000 --> 00:40:25,000 zero. I should probably tell you a 456 00:40:25,000 --> 00:40:29,000 little bit about what the curl is, what it measures and what it 457 00:40:29,000 --> 00:40:34,000 does because that is something that is probably useful. 458 00:40:34,000 --> 00:40:37,000 It is a very strange quantity if you put it in that form. 459 00:40:37,000 --> 00:40:42,000 Yes? I think it is the same as the 460 00:40:42,000 --> 00:40:45,000 physics one, but I haven't checked the physics textbook. 461 00:40:45,000 --> 00:40:49,000 I believe it is the same. Yes, I think it is the same as 462 00:40:49,000 --> 00:40:53,000 the physics one. It is not the opposite this 463 00:40:53,000 --> 00:40:55,000 time. Of course, in physics maybe you 464 00:40:55,000 --> 00:40:59,000 have seen curl in space. We are going to see curl in 465 00:40:59,000 --> 00:41:07,000 space in two or three weeks. Yes? 466 00:41:07,000 --> 00:41:11,000 Yes. Well, you can also use it. If you fail this test then you 467 00:41:11,000 --> 00:41:14,000 know for sure that you are not gradient field so you might as 468 00:41:14,000 --> 00:41:18,000 well do that. If you satisfy the test but you 469 00:41:18,000 --> 00:41:24,000 are not defined everywhere then there is still a bit of 470 00:41:24,000 --> 00:41:29,000 ambiguity and you don't know for sure. 471 00:41:29,000 --> 00:41:40,000 OK. Let's try to see a little bit 472 00:41:40,000 --> 00:41:48,000 what the curl measures. Just to give you some 473 00:41:48,000 --> 00:41:55,000 intuition, let's first think about a velocity field. 474 00:41:55,000 --> 00:42:10,000 The curl measures the rotation component of a motion. 475 00:42:10,000 --> 00:42:13,000 If you want a fancy word, it measures the vorticity of a 476 00:42:13,000 --> 00:42:16,000 motion. It tells you how much twisting 477 00:42:16,000 --> 00:42:19,000 is taking place at a given point. 478 00:42:19,000 --> 00:42:24,000 For example, if I take a constant vector 479 00:42:24,000 --> 00:42:32,000 field where my fluid is just all moving in the same direction 480 00:42:32,000 --> 00:42:37,000 where this is just constants then, 481 00:42:37,000 --> 00:42:41,000 of course, the curl is zero. Because if you take the 482 00:42:41,000 --> 00:42:43,000 partials you get zero. And, indeed, 483 00:42:43,000 --> 00:42:46,000 that is not what you would call swirling. 484 00:42:46,000 --> 00:42:58,000 There is no vortex in here. Let's do another one where this 485 00:42:58,000 --> 00:43:02,000 is still nothing going on. Let's say that I take the 486 00:43:02,000 --> 00:43:06,000 radial vector field where everything just flows away from 487 00:43:06,000 --> 00:43:11,000 the origin. That is f equals x, y. 488 00:43:11,000 --> 00:43:16,000 Well, if I take the curl, I have to take partial over 489 00:43:16,000 --> 00:43:18,000 partial x of the second component, 490 00:43:18,000 --> 00:43:21,000 which is y, minus partial over partial y of 491 00:43:21,000 --> 00:43:22,000 the first component, which is x. 492 00:43:22,000 --> 00:43:25,000 I will get zero. And, indeed, 493 00:43:25,000 --> 00:43:29,000 if you think about what is going on here, 494 00:43:29,000 --> 00:43:32,000 there is no rotation involved. On the other hand, 495 00:43:32,000 --> 00:43:45,000 if you consider our favorite rotation vector field -- -- 496 00:43:45,000 --> 00:44:00,000 negative y and x then this curl is going to be N sub x minus M 497 00:44:00,000 --> 00:44:08,000 sub y, one plus one equals two. 498 00:44:08,000 --> 00:44:13,000 That corresponds to the fact that we are rotating. 499 00:44:13,000 --> 00:44:16,000 Actually, we are rotating at unit angular speed. 500 00:44:16,000 --> 00:44:20,000 The curl actually measures twice the angular speed of a 501 00:44:20,000 --> 00:44:24,000 rotation part of a motion at any given point. 502 00:44:24,000 --> 00:44:26,000 Now, if you have an actual motion, 503 00:44:26,000 --> 00:44:30,000 a more complicated field than these then no matter where you 504 00:44:30,000 --> 00:44:34,000 are you can think of a motion as a combination of translation 505 00:44:34,000 --> 00:44:37,000 effects, maybe dilation effects, 506 00:44:37,000 --> 00:44:43,000 maybe rotation effects, possibly other things like that. 507 00:44:43,000 --> 00:44:48,000 And what a curl will measure is how intense the rotation effect 508 00:44:48,000 --> 00:44:52,000 is at that particular point. I am not going to try to make a 509 00:44:52,000 --> 00:44:55,000 much more precise statement. A precise statement is what a 510 00:44:55,000 --> 00:44:58,000 curl measures is really this quantity up there. 511 00:44:58,000 --> 00:45:01,000 But the intuition you should have is it measures how much 512 00:45:01,000 --> 00:45:04,000 rotation is taking place at any given point. 513 00:45:04,000 --> 00:45:06,000 And, of course, in a complicated motion you 514 00:45:06,000 --> 00:45:09,000 might have more rotation at some point than at some others, 515 00:45:09,000 --> 00:45:12,000 which is why the curl will depend on x and y. 516 00:45:12,000 --> 00:45:20,000 It is not just a constant because how much you rotate 517 00:45:20,000 --> 00:45:26,000 depends on where you are. If you are looking at actual 518 00:45:26,000 --> 00:45:30,000 wind velocities in weather prediction then the regions with 519 00:45:30,000 --> 00:45:33,000 high curl tend to be hurricanes or tornadoes or things like 520 00:45:33,000 --> 00:45:37,000 that. They are not very pleasant 521 00:45:37,000 --> 00:45:40,000 things. And the sign of a curl tells 522 00:45:40,000 --> 00:45:43,000 you whether you are going clockwise or counterclockwise. 523 00:45:43,000 --> 00:46:09,000 524 00:46:09,000 --> 00:46:27,000 Curl measures twice the angular velocity of the rotation 525 00:46:27,000 --> 00:46:41,000 component of a velocity field. Now, what about a force field? 526 00:46:41,000 --> 00:46:44,000 Because, after all, how we got to this was coming 527 00:46:44,000 --> 00:46:47,000 from and trying to understand forces and the work they do. 528 00:46:47,000 --> 00:46:50,000 So I should tell you what it means for a force. 529 00:46:50,000 --> 00:47:10,000 Well, the curl of a force field -- -- measures the torque 530 00:47:10,000 --> 00:47:29,000 exerted on a test object that you put at any point. 531 00:47:29,000 --> 00:47:36,000 Remember, torque is the rotational analog of the force. 532 00:47:36,000 --> 00:47:41,000 We had this analogy about velocity versus angular velocity 533 00:47:41,000 --> 00:47:45,000 and mass versus moment of inertia. 534 00:47:45,000 --> 00:47:49,000 And then, in that analogy, force divided by the mass is 535 00:47:49,000 --> 00:47:53,000 what will cause acceleration, which is the derivative of 536 00:47:53,000 --> 00:47:56,000 velocity. Torque divided by moment of 537 00:47:56,000 --> 00:47:59,000 inertia is what will cause the angular acceleration, 538 00:47:59,000 --> 00:48:02,000 namely the derivative of angular velocity. 539 00:48:02,000 --> 00:48:04,000 Maybe I should write that down. 540 00:48:04,000 --> 00:48:18,000 541 00:48:18,000 --> 00:48:31,000 Torque divided by moment of inertia is going to be d over dt 542 00:48:31,000 --> 00:48:38,000 of angular velocity. I leave it up to your physics 543 00:48:38,000 --> 00:48:41,000 teachers to decide what letters to use for all these things. 544 00:48:41,000 --> 00:48:49,000 That is the analog of force divided by mass equals 545 00:48:49,000 --> 00:48:56,000 acceleration, which is d over dt of velocity. 546 00:48:56,000 --> 00:49:03,000 And so now you see if the curl of a velocity field measure the 547 00:49:03,000 --> 00:49:07,000 angular velocity of its rotation then, 548 00:49:07,000 --> 00:49:13,000 by this analogy, the curl of a force field 549 00:49:13,000 --> 00:49:24,000 should measure the torque it exerts on a mass per unit moment 550 00:49:24,000 --> 00:49:28,000 of inertia. Concretely, if you imagine that 551 00:49:28,000 --> 00:49:29,000 you are putting something in there, 552 00:49:29,000 --> 00:49:32,000 you know, if you are in a velocity field the curl will 553 00:49:32,000 --> 00:49:35,000 tell you how fast your guy is spinning at a given time. 554 00:49:35,000 --> 00:49:37,000 If you put something that floats, for example, 555 00:49:37,000 --> 00:49:40,000 in your fluid, something very light then it is 556 00:49:40,000 --> 00:49:44,000 going to start spinning. And the curl of a velocity 557 00:49:44,000 --> 00:49:48,000 field tells you how fast it is spinning at any given time up to 558 00:49:48,000 --> 00:49:51,000 a factor of two. And the curl of a force field 559 00:49:51,000 --> 00:49:55,000 tells you how quickly the angular velocity is going to 560 00:49:55,000 --> 00:50:01,000 increase or decrease. OK. 561 00:50:01,000 --> 00:50:04,000 Well, next time we are going to see Green's theorem which is 562 00:50:04,000 --> 00:50:08,000 actually going to tell us a lot more about curl and failure of 563 00:50:08,000 --> 00:50:11,000 conservativeness. 564 00:50:11,000 --> 00:50:16,000