1 00:00:06,915 --> 00:00:07,540 JOEL LEWIS: Hi. 2 00:00:07,540 --> 00:00:09,330 Welcome back to recitation. 3 00:00:09,330 --> 00:00:12,510 I've got a nice exercise here on Stokes' Theorem for you. 4 00:00:12,510 --> 00:00:14,380 Now this problem is a little bit more 5 00:00:14,380 --> 00:00:16,110 sophisticated than a lot of problems 6 00:00:16,110 --> 00:00:17,630 we've been doing in recitation. 7 00:00:17,630 --> 00:00:20,400 So it requires a little bit more thought, and it also 8 00:00:20,400 --> 00:00:22,940 involves more mathematical sophistication. 9 00:00:22,940 --> 00:00:28,230 So we're doing a clever kind of proof here that I like. 10 00:00:28,230 --> 00:00:31,170 So hopefully you'll like this one. 11 00:00:31,170 --> 00:00:33,670 It's a little bit in a different style than some of the ones 12 00:00:33,670 --> 00:00:35,600 we've done. 13 00:00:35,600 --> 00:00:38,000 So I think I need to talk about it a little bit before we 14 00:00:38,000 --> 00:00:38,630 get started. 15 00:00:38,630 --> 00:00:41,580 So let's let F be the field [x, y, z]. 16 00:00:41,580 --> 00:00:45,870 So this is our radial field that we've seen a lot in recitation. 17 00:00:45,870 --> 00:00:47,840 So what I'd like you to do is prove 18 00:00:47,840 --> 00:00:52,200 that this field is not the curl of any field G. All right. 19 00:00:52,200 --> 00:00:53,910 So I'd like you to show that there's 20 00:00:53,910 --> 00:00:58,790 no field G such that F is equal to the curl of G. 21 00:00:58,790 --> 00:01:02,370 Now, rather than just saying that to you 22 00:01:02,370 --> 00:01:04,750 and letting you run off, I have a suggestion 23 00:01:04,750 --> 00:01:07,750 for an interesting way you could go about this. 24 00:01:07,750 --> 00:01:11,340 And this interesting way is going to use Stokes' Theorem. 25 00:01:11,340 --> 00:01:16,400 So what I'd like you to do is a proof by contradiction. 26 00:01:16,400 --> 00:01:18,430 OK, so what you're going to do is 27 00:01:18,430 --> 00:01:21,740 you're going to assume that F is a curl. 28 00:01:21,740 --> 00:01:22,240 OK? 29 00:01:22,240 --> 00:01:23,890 So you're going to assume that there 30 00:01:23,890 --> 00:01:28,920 is some G such that F is curl G. And then you're 31 00:01:28,920 --> 00:01:33,890 going to use that to get a ridiculous conclusion. 32 00:01:33,890 --> 00:01:35,760 So you're going to start with that premise, 33 00:01:35,760 --> 00:01:38,960 and you're going to end up with a contradiction. 34 00:01:38,960 --> 00:01:43,540 So these two arrows colliding into each other 35 00:01:43,540 --> 00:01:46,810 is a symbol that mathematicians use for a contradiction. 36 00:01:46,810 --> 00:01:48,640 So you're going to start from this premise, 37 00:01:48,640 --> 00:01:50,460 and you're going to reach a contradiction. 38 00:01:50,460 --> 00:01:52,750 And what that's going to show is that your premise 39 00:01:52,750 --> 00:01:54,071 couldn't be right. 40 00:01:54,071 --> 00:01:54,570 Yeah? 41 00:01:54,570 --> 00:01:57,260 Because if you start from a true premise, 42 00:01:57,260 --> 00:02:00,050 well then all your conclusions should be true as well. 43 00:02:00,050 --> 00:02:02,610 So if you reach a false conclusion, 44 00:02:02,610 --> 00:02:05,470 then you must have had a false premise. 45 00:02:05,470 --> 00:02:07,950 So what you're going to do is you're 46 00:02:07,950 --> 00:02:11,590 going to take a sphere of radius b centered at the origin. 47 00:02:11,590 --> 00:02:13,750 And a curve C on the sphere. 48 00:02:13,750 --> 00:02:16,640 You know, a simple, closed curve. 49 00:02:16,640 --> 00:02:21,730 So assuming that F is this curl of G, what I'd like you to do 50 00:02:21,730 --> 00:02:25,330 is use Stokes' Theorem to interpret 51 00:02:25,330 --> 00:02:31,170 the line integral of G dot dr over C in two different ways. 52 00:02:31,170 --> 00:02:32,110 OK? 53 00:02:32,110 --> 00:02:35,520 And interpreting this line integral in two different ways, 54 00:02:35,520 --> 00:02:37,130 you're going to reach a contradiction, 55 00:02:37,130 --> 00:02:40,130 and that will show that F really isn't a curl. 56 00:02:40,130 --> 00:02:41,760 So that's what I'd like you to do. 57 00:02:41,760 --> 00:02:43,460 So why don't you pause the video, 58 00:02:43,460 --> 00:02:45,966 go ahead and see if you can work that out, come back, 59 00:02:45,966 --> 00:02:47,340 and we'll talk about it together. 60 00:02:55,660 --> 00:02:57,910 I hope you enjoyed working on this problem. 61 00:02:57,910 --> 00:02:59,320 Let's get started on it. 62 00:02:59,320 --> 00:03:03,940 So as I was saying before we started, what we're going to do 63 00:03:03,940 --> 00:03:06,280 is we're looking for a proof by contradiction. 64 00:03:06,280 --> 00:03:08,738 So as the problem says, we're going to start with a sphere. 65 00:03:11,520 --> 00:03:14,170 And I'm going to take this curve C-- 66 00:03:14,170 --> 00:03:17,940 some simply connected closed curve that's going to go around 67 00:03:17,940 --> 00:03:21,354 the back of the sphere, and it's going to be oriented 68 00:03:21,354 --> 00:03:22,895 one way or the other-- and it's going 69 00:03:22,895 --> 00:03:25,580 to divide this sphere into two pieces. 70 00:03:25,580 --> 00:03:30,050 So there's the one cap on one side of it, S_1. 71 00:03:30,050 --> 00:03:33,380 And then there's-- whatever the other piece on the other side 72 00:03:33,380 --> 00:03:35,430 of it is, S_2. 73 00:03:35,430 --> 00:03:36,030 OK. 74 00:03:36,030 --> 00:03:38,720 And so what we're going to do is we're going to think 75 00:03:38,720 --> 00:03:40,921 about, what is this line integral? 76 00:03:40,921 --> 00:03:41,420 OK. 77 00:03:41,420 --> 00:03:45,340 So this is our curve C here on the sphere. 78 00:03:45,340 --> 00:03:54,400 So the integral over C of G dot dr. 79 00:03:54,400 --> 00:03:56,910 So this is what the problem suggests we think about. 80 00:03:56,910 --> 00:03:59,780 So this is a line integral of a field dot dr 81 00:03:59,780 --> 00:04:01,450 over the boundary of a surface. 82 00:04:01,450 --> 00:04:03,640 Well, actually, it's the boundary of two surfaces. 83 00:04:03,640 --> 00:04:04,140 Right? 84 00:04:08,440 --> 00:04:11,610 C is the boundary of S1, and C-- if we orient it 85 00:04:11,610 --> 00:04:15,120 the other way-- is the boundary of S_2, when 86 00:04:15,120 --> 00:04:18,730 we orient them both outwards. 87 00:04:18,730 --> 00:04:21,660 OK, so what is this? 88 00:04:21,660 --> 00:04:23,429 So Stokes' Theorem tells us something 89 00:04:23,429 --> 00:04:24,470 about this line integral. 90 00:04:24,470 --> 00:04:28,500 So let's first think about this as the top cap-- that 91 00:04:28,500 --> 00:04:32,770 cap S1-- with boundary C oriented so that they 92 00:04:32,770 --> 00:04:33,720 agree with each other. 93 00:04:33,720 --> 00:04:36,210 So the normal is outwards on the sphere, 94 00:04:36,210 --> 00:04:38,730 and C is proceeding in the direction 95 00:04:38,730 --> 00:04:40,390 that I've drawn the arrow here. 96 00:04:40,390 --> 00:04:43,140 Well, in that circumstance, we have 97 00:04:43,140 --> 00:04:48,600 that the integral around C of G dot dr, by Stokes' Theorem, 98 00:04:48,600 --> 00:05:00,950 is equal to the surface integral over S_1 of curl of G dot 99 00:05:00,950 --> 00:05:04,710 n with respect to surface area. 100 00:05:04,710 --> 00:05:05,210 Right? 101 00:05:05,210 --> 00:05:06,290 So this is just Stokes' Theorem. 102 00:05:06,290 --> 00:05:08,020 Stokes' Theorem says the line integral 103 00:05:08,020 --> 00:05:11,680 of G around the boundary curve is equal to the surface 104 00:05:11,680 --> 00:05:14,980 integral of the curl of G over the region, 105 00:05:14,980 --> 00:05:17,370 provided all of our orientations are correct. 106 00:05:17,370 --> 00:05:17,870 OK. 107 00:05:17,870 --> 00:05:20,820 Well, we know though what curl of G 108 00:05:20,820 --> 00:05:24,820 is, because by assumption, F is equal to curl of G. OK, 109 00:05:24,820 --> 00:05:36,050 so this is equal to the surface integral over S1 of F dot n dS. 110 00:05:36,050 --> 00:05:38,300 So in the first step, we use Stokes' Theorem. 111 00:05:38,300 --> 00:05:41,090 In the second step, we use our assumption 112 00:05:41,090 --> 00:05:45,410 that curl G is equal to F. Well, now what? 113 00:05:45,410 --> 00:05:46,550 But we know what F is. 114 00:05:46,550 --> 00:05:47,050 Right? 115 00:05:47,050 --> 00:05:49,430 F is this radial field [x, y, z]. 116 00:05:49,430 --> 00:05:52,760 So F and n are pointing in the same direction. 117 00:05:52,760 --> 00:05:55,380 They're parallel to each other. n is a unit vector, 118 00:05:55,380 --> 00:05:58,290 so this is just the length of F. This F dot 119 00:05:58,290 --> 00:06:00,360 n is just the length of F. And since we're 120 00:06:00,360 --> 00:06:04,310 on a sphere of radius b, this is just b. 121 00:06:04,310 --> 00:06:06,680 OK, so the integrand is just b. 122 00:06:06,680 --> 00:06:14,530 So this is the integral over S_1 of b 123 00:06:14,530 --> 00:06:23,580 dS, which is b times the area of S_1. 124 00:06:23,580 --> 00:06:24,930 OK. 125 00:06:24,930 --> 00:06:27,440 One thing I'd like you to notice is that in particular, this 126 00:06:27,440 --> 00:06:29,250 is a positive number. 127 00:06:29,250 --> 00:06:32,340 b is positive and the area is positive. 128 00:06:32,340 --> 00:06:32,840 OK. 129 00:06:32,840 --> 00:06:35,150 So that's our first interpretation. 130 00:06:35,150 --> 00:06:40,910 So we took our field G that we suppose exists, 131 00:06:40,910 --> 00:06:44,260 and we integrate it around this curve C, 132 00:06:44,260 --> 00:06:47,130 and we apply Stokes' Theorem, and then 133 00:06:47,130 --> 00:06:49,720 the fact that we know what F is means 134 00:06:49,720 --> 00:06:54,290 that we know what F dot n is, and so that makes our surface 135 00:06:54,290 --> 00:06:55,860 integral very easy to compute. 136 00:06:55,860 --> 00:06:59,330 And it turns out to be b times the area of S, which, 137 00:06:59,330 --> 00:07:03,360 I just happened to notice, is a positive number. 138 00:07:03,360 --> 00:07:04,140 OK. 139 00:07:04,140 --> 00:07:06,790 Well, now we can do the same trick 140 00:07:06,790 --> 00:07:08,981 on the other half of the sphere. 141 00:07:08,981 --> 00:07:09,480 Right? 142 00:07:09,480 --> 00:07:11,160 So we just did the top cap here. 143 00:07:11,160 --> 00:07:12,100 We did S_1. 144 00:07:12,100 --> 00:07:14,920 So now we have the bottom cap, or whatever. 145 00:07:14,920 --> 00:07:17,951 All the rest of the sphere, S_2. 146 00:07:17,951 --> 00:07:18,450 OK. 147 00:07:18,450 --> 00:07:23,050 So we can also get that the integral over C of G dot dr, 148 00:07:23,050 --> 00:07:25,200 we can interpret it in terms of Stokes' Theorem. 149 00:07:25,200 --> 00:07:28,224 But notice then that C-- we still 150 00:07:28,224 --> 00:07:29,390 want to use the same normal. 151 00:07:29,390 --> 00:07:31,140 We like outwards pointing normals. 152 00:07:31,140 --> 00:07:34,450 So we're going to have to orient C the other way in order 153 00:07:34,450 --> 00:07:36,480 to make Stokes' Theorem make sense. 154 00:07:36,480 --> 00:07:40,619 So let's walk over here where we have some empty board space. 155 00:07:40,619 --> 00:07:42,160 So we want to orient C the other way. 156 00:07:42,160 --> 00:07:44,810 So in other words, we're going to take the negative 157 00:07:44,810 --> 00:07:47,070 of this line integral. 158 00:07:47,070 --> 00:07:51,950 So it's minus G dot dr. And if we 159 00:07:51,950 --> 00:07:54,980 apply Stokes' Theorem to this line integral-- so this 160 00:07:54,980 --> 00:07:58,460 is the same line integral, but with the opposite orientation 161 00:07:58,460 --> 00:08:03,690 on C and so with the opposite sign-- by Stokes' Theorem, 162 00:08:03,690 --> 00:08:14,470 this is equal to the integral over S_2 of the curl of G 163 00:08:14,470 --> 00:08:16,520 dot n dS. 164 00:08:16,520 --> 00:08:18,830 And so the way I've set this up, this is still 165 00:08:18,830 --> 00:08:20,730 my outward-pointing normal. 166 00:08:20,730 --> 00:08:21,550 OK. 167 00:08:21,550 --> 00:08:23,950 But again, we can use our assumption 168 00:08:23,950 --> 00:08:29,220 and we can get curl of G is equal to F, 169 00:08:29,220 --> 00:08:31,760 because we're assuming that G has this property. 170 00:08:31,760 --> 00:08:38,745 So this is equal to the integral over S_2 171 00:08:38,745 --> 00:08:44,340 of F dot n with respect to surface area. 172 00:08:44,340 --> 00:08:46,360 And again, n is the outward-pointing normal, 173 00:08:46,360 --> 00:08:50,350 and F is parallel to it. 174 00:08:50,350 --> 00:08:52,090 So this dot product is just the length 175 00:08:52,090 --> 00:08:54,930 of F. The outward pointing unit normal. 176 00:08:54,930 --> 00:08:57,050 So this is just the length of F, which is b. 177 00:08:57,050 --> 00:09:04,120 So this is equal to b times the area of S_2, 178 00:09:04,120 --> 00:09:06,880 which is also positive. 179 00:09:06,880 --> 00:09:08,730 So what have we just shown? 180 00:09:08,730 --> 00:09:10,970 Well, we started from the assumption 181 00:09:10,970 --> 00:09:14,840 that there exists a G such that F is the curl of G. 182 00:09:14,840 --> 00:09:17,480 And starting from that assumption-- 183 00:09:17,480 --> 00:09:22,960 let's look-- we showed that the line integral around C of G 184 00:09:22,960 --> 00:09:29,170 dot dr is equal to some positive number. 185 00:09:29,170 --> 00:09:33,990 And we also showed, over here, that the negative 186 00:09:33,990 --> 00:09:37,220 of the line integral of G around C 187 00:09:37,220 --> 00:09:39,300 is equal to some positive number. 188 00:09:39,300 --> 00:09:40,700 Well, this is clearly absurd. 189 00:09:40,700 --> 00:09:42,420 That can't be true. 190 00:09:42,420 --> 00:09:46,480 So starting from our assumption that F was the curl of G-- 191 00:09:46,480 --> 00:09:48,990 that there is a G such the F is the curl of G-- 192 00:09:48,990 --> 00:09:50,590 we reached an absurd conclusion. 193 00:09:50,590 --> 00:09:52,610 We reached a conclusion that the same number 194 00:09:52,610 --> 00:09:55,210 is both positive and negative. 195 00:09:55,210 --> 00:09:56,280 But that can't happen. 196 00:09:56,280 --> 00:09:58,730 So that means our premise had to be false. 197 00:09:58,730 --> 00:09:59,360 OK. 198 00:09:59,360 --> 00:10:00,485 So this is a contradiction. 199 00:10:22,284 --> 00:10:24,230 So our assumption is false. 200 00:10:24,230 --> 00:10:25,940 And our assumption was-- that we used 201 00:10:25,940 --> 00:10:28,180 to get this whole thing started-- that F was 202 00:10:28,180 --> 00:10:31,170 the curl of some G. All right. 203 00:10:31,170 --> 00:10:32,310 So what have we shown? 204 00:10:32,310 --> 00:10:35,340 So we used a nice argument here with Stokes' Theorem 205 00:10:35,340 --> 00:10:38,360 in order to show that certain fields aren't 206 00:10:38,360 --> 00:10:39,930 the curl of other fields. 207 00:10:39,930 --> 00:10:43,090 So Stokes' Theorem limits the kind of fields that 208 00:10:43,090 --> 00:10:44,630 can be curls of other fields. 209 00:10:44,630 --> 00:10:48,060 Now perhaps, you may have thought of other theorems 210 00:10:48,060 --> 00:10:51,210 that you can use that also limit what sorts of fields 211 00:10:51,210 --> 00:10:51,930 can be curls. 212 00:10:51,930 --> 00:10:55,090 And so there are other ways to reach this true conclusion 213 00:10:55,090 --> 00:10:58,250 that our field F-- whose components are x, y, and z-- 214 00:10:58,250 --> 00:11:00,310 is not the curl of any field. 215 00:11:00,310 --> 00:11:02,700 This isn't the only way to reach that conclusion. 216 00:11:02,700 --> 00:11:06,000 But this is a nice way that shows that Stokes' Theorem puts 217 00:11:06,000 --> 00:11:08,970 some limitations on what fields can behave like if they're 218 00:11:08,970 --> 00:11:10,380 going to be curls. 219 00:11:10,380 --> 00:11:11,972 I'll stop there.