1 00:00:00,000 --> 00:00:00,890 2 00:00:00,890 --> 00:00:05,729 Let's say I have the vector field v. 3 00:00:05,730 --> 00:00:09,920 The vector at any given point is equal to minus-- or the 4 00:00:09,919 --> 00:00:14,570 magnitude of my x direction is actually dependent on y. 5 00:00:14,570 --> 00:00:19,280 So where we are in the y coordinate and the xy plane. 6 00:00:19,280 --> 00:00:21,385 Plus-- and then my magnitude in the y direction 7 00:00:21,385 --> 00:00:22,470 is dependent on x. 8 00:00:22,469 --> 00:00:25,689 9 00:00:25,690 --> 00:00:26,920 Fair enough. 10 00:00:26,920 --> 00:00:29,210 So first, let's just chuck through this, and 11 00:00:29,210 --> 00:00:31,089 figure out it's curl. 12 00:00:31,089 --> 00:00:32,679 So the curl of v. 13 00:00:32,679 --> 00:00:36,450 14 00:00:36,450 --> 00:00:43,160 It is equal to our del vector operator, cross v. 15 00:00:43,159 --> 00:00:45,779 Which is nothing but this. 16 00:00:45,780 --> 00:00:49,200 And even though this looks like a two dimensional vector field, 17 00:00:49,200 --> 00:00:51,480 we actually have to take the cross product in 3 dimensions. 18 00:00:51,479 --> 00:00:55,419 Because a curl is just like torque, and when you-- like we 19 00:00:55,420 --> 00:00:57,410 did the right hand rule when we studied-- well, I hope you 20 00:00:57,409 --> 00:01:00,750 watched some of the videos on magnetism and torque-- but the 21 00:01:00,750 --> 00:01:03,780 torque actually goes in a direction that is perpendicular 22 00:01:03,780 --> 00:01:08,060 to both of the vectors in your cross product. 23 00:01:08,060 --> 00:01:11,549 If both of these only have x and y components, your actual 24 00:01:11,549 --> 00:01:13,640 result is going to be in the z direction. 25 00:01:13,640 --> 00:01:16,209 It's going to be perpendicular to both of these vectors. 26 00:01:16,209 --> 00:01:18,750 So when you take the cross product, you still have to 27 00:01:18,750 --> 00:01:20,730 do it in 3 dimensions. 28 00:01:20,730 --> 00:01:31,659 So i j k, partial with respect to x, partial with respect to 29 00:01:31,659 --> 00:01:35,399 y, partial with respect to z. 30 00:01:35,400 --> 00:01:38,690 The x component is minus y. 31 00:01:38,689 --> 00:01:40,500 The j component is x. 32 00:01:40,500 --> 00:01:42,920 And we have no k component. 33 00:01:42,920 --> 00:01:45,109 This one should be a little bit cleaner to calculate 34 00:01:45,109 --> 00:01:47,370 than the last example. 35 00:01:47,370 --> 00:01:50,150 So this is going to be equal to-- well, see the i 36 00:01:50,150 --> 00:01:53,195 component-- let's cross out its column and its row. 37 00:01:53,194 --> 00:02:00,179 So it's going to be the partial with respect to y of 0 38 00:02:00,180 --> 00:02:06,980 minus the partial with respect to z of x. 39 00:02:06,980 --> 00:02:08,030 Minus that. 40 00:02:08,030 --> 00:02:13,140 And all of that times i minus-- and now it's going 41 00:02:13,139 --> 00:02:13,869 to be the j component. 42 00:02:13,870 --> 00:02:17,180 Remember, you do plus minus plus. 43 00:02:17,180 --> 00:02:18,960 So minus and this is the j component. 44 00:02:18,960 --> 00:02:20,879 Cross out its row and column, the partial derivative 45 00:02:20,879 --> 00:02:23,299 of x with respect to 0. 46 00:02:23,300 --> 00:02:26,550 Or partial derivative of 0 with respect to x, actually. 47 00:02:26,550 --> 00:02:32,450 Minus the partial derivative of z. 48 00:02:32,449 --> 00:02:33,839 Or the partial derivative of the [? vector ?] 49 00:02:33,840 --> 00:02:37,539 z of minus y. 50 00:02:37,539 --> 00:02:39,859 That's its j component. 51 00:02:39,860 --> 00:02:41,820 And finally, plus its k component. 52 00:02:41,819 --> 00:02:42,629 Row and column. 53 00:02:42,629 --> 00:02:50,740 So the partial derivative with respect to x minus the partial 54 00:02:50,740 --> 00:02:55,280 derivative of y with respect to minus y. 55 00:02:55,280 --> 00:02:57,360 And I know you can't read it, but that's that minus y there, 56 00:02:57,360 --> 00:02:59,900 and that's the k component. 57 00:02:59,900 --> 00:03:00,990 And now let's simplify it. 58 00:03:00,990 --> 00:03:03,810 I'll simplify it above it. 59 00:03:03,810 --> 00:03:06,219 So this term-- let's see, partial derivative of 0. 60 00:03:06,219 --> 00:03:07,949 Well, that's 0. 61 00:03:07,949 --> 00:03:10,359 Partial derivative of x with respect to z. 62 00:03:10,360 --> 00:03:14,310 Well, as far as z's concerned, x is a constant, so that's 0. 63 00:03:14,310 --> 00:03:17,280 Partial derivative of 0 with respect to x, 0. 64 00:03:17,280 --> 00:03:20,550 Partial derivative of minus y with respect to z. 65 00:03:20,550 --> 00:03:22,740 As far as z's concerned, y is also constant. 66 00:03:22,740 --> 00:03:24,010 So that's 0. 67 00:03:24,009 --> 00:03:25,530 So all we're left is with this last term. 68 00:03:25,530 --> 00:03:26,270 That was pretty straightforward. 69 00:03:26,270 --> 00:03:28,320 Why don't we just cross all of that out? 70 00:03:28,319 --> 00:03:30,419 And that makes intuitive sense, too, right? 71 00:03:30,419 --> 00:03:37,000 Because at least from x's point of view, the rotation is-- 72 00:03:37,000 --> 00:03:39,990 although, the rotation, if you think about it, is going to be 73 00:03:39,990 --> 00:03:47,080 in the direction perpendicular to both the x direction and 74 00:03:47,080 --> 00:03:51,380 perpendicular to the direction in which it is changing. 75 00:03:51,379 --> 00:03:54,370 You could kind of view it orthogonally, to its direction 76 00:03:54,370 --> 00:03:55,650 of motion, which is y. 77 00:03:55,650 --> 00:03:57,310 So it would be the z direction. 78 00:03:57,310 --> 00:03:58,860 If that confuses you, don't worry about it. 79 00:03:58,860 --> 00:04:01,420 But if it doesn't, then you could apply the same 80 00:04:01,419 --> 00:04:03,349 argument to the j vector. 81 00:04:03,349 --> 00:04:05,269 But anyway, lets simplify this. 82 00:04:05,270 --> 00:04:07,890 So this is equal to the partial derivative of 83 00:04:07,889 --> 00:04:09,279 x with respect to x. 84 00:04:09,280 --> 00:04:11,509 Well, that's just 1. 85 00:04:11,509 --> 00:04:15,269 Minus the partial derivative of y-- of minus y 86 00:04:15,270 --> 00:04:16,129 with respect to y. 87 00:04:16,129 --> 00:04:18,750 Well, that's just minus 1. 88 00:04:18,750 --> 00:04:20,560 So it equals 2. 89 00:04:20,560 --> 00:04:26,329 So the curl, at any point of this vector field, is 2. 90 00:04:26,329 --> 00:04:28,379 Let's see what this vector field looks like, and let's 91 00:04:28,379 --> 00:04:33,589 see if that gives us-- if our intuition holds 92 00:04:33,589 --> 00:04:34,609 in this example. 93 00:04:34,610 --> 00:04:38,360 And let me try to make it a little bit bigger. 94 00:04:38,360 --> 00:04:41,650 Make the window bigger. 95 00:04:41,649 --> 00:04:42,509 There you go. 96 00:04:42,509 --> 00:04:45,149 Well, I think it's clear, you know, right when you look at 97 00:04:45,149 --> 00:04:47,009 it, that this vector field looks like it's spinning. 98 00:04:47,009 --> 00:04:49,759 If you were to stick something, especially in the middle, it's 99 00:04:49,759 --> 00:04:51,449 very clear that it would spin. 100 00:04:51,449 --> 00:04:53,449 But what might be a little unintuitive-- you might think, 101 00:04:53,449 --> 00:04:55,870 wow, well, wouldn't something spin faster near the center 102 00:04:55,870 --> 00:04:57,280 than it would here? 103 00:04:57,279 --> 00:04:59,659 Why is the-- you know, the curl we got is 2. 104 00:04:59,660 --> 00:05:00,450 It's a constant. 105 00:05:00,449 --> 00:05:04,149 The curl is the same throughout this entire vector field. 106 00:05:04,149 --> 00:05:06,679 So you'll be like, whoa, that's kind of implying that the field 107 00:05:06,680 --> 00:05:09,439 is making something spin equally, no matter where 108 00:05:09,439 --> 00:05:10,209 you are in the field. 109 00:05:10,209 --> 00:05:11,949 Let's see if that makes sense. 110 00:05:11,949 --> 00:05:13,729 Well, in the middle, it definitely makes sense that 111 00:05:13,730 --> 00:05:14,970 something is spinning. 112 00:05:14,970 --> 00:05:17,360 If I had a little stick here, I'd be pushing in this 113 00:05:17,360 --> 00:05:19,889 direction, with not that much of a magnitude in 114 00:05:19,889 --> 00:05:20,789 that direction. 115 00:05:20,790 --> 00:05:23,720 And then I'd be pushing down to the right in this direction, 116 00:05:23,720 --> 00:05:25,570 so it would cause it to spin. 117 00:05:25,569 --> 00:05:29,899 But what if I had that same stick here? 118 00:05:29,899 --> 00:05:31,810 You'd say, well, on the top right I'm pushing 119 00:05:31,810 --> 00:05:34,019 up, up and to the left. 120 00:05:34,019 --> 00:05:37,180 And the bottom left, I'm also pushing up and to the left. 121 00:05:37,180 --> 00:05:38,269 You know, it wouldn't spin as much. 122 00:05:38,269 --> 00:05:38,879 But it would. 123 00:05:38,879 --> 00:05:42,389 Because the difference in magnitudes of these 2-- you 124 00:05:42,389 --> 00:05:45,529 could almost view them as the torque producing forces-- the 125 00:05:45,529 --> 00:05:49,389 difference in magnitude is enough that you'd still have 126 00:05:49,389 --> 00:05:53,029 the same counterclockwise rotation here as you 127 00:05:53,029 --> 00:05:53,799 would have here. 128 00:05:53,800 --> 00:05:58,110 So because the curl is a constant positive number, when 129 00:05:58,110 --> 00:06:01,410 we look at the xy plane like this, if you put a twig-- that 130 00:06:01,410 --> 00:06:04,390 same twig anywhere where you put it on this plane-- 131 00:06:04,389 --> 00:06:07,370 it'll have the same counterclockwise rotation. 132 00:06:07,370 --> 00:06:09,370 I think that's pretty neat. 133 00:06:09,370 --> 00:06:12,149 Now let's do a little experiment. 134 00:06:12,149 --> 00:06:18,599 What would have happened if this was plus y. 135 00:06:18,600 --> 00:06:20,900 So let's just do that experiment. 136 00:06:20,899 --> 00:06:29,929 If this was plus y then this would have been plus y, then 137 00:06:29,930 --> 00:06:32,060 this final term-- we would have taken the partial derivative 138 00:06:32,060 --> 00:06:35,110 with respect to y of plus y-- and so this would have 139 00:06:35,110 --> 00:06:37,259 been 1 minus plus 1. 140 00:06:37,259 --> 00:06:40,050 And then our curl would have been 0. 141 00:06:40,050 --> 00:06:42,710 Which would have meant that we would have had no rotation. 142 00:06:42,709 --> 00:06:44,029 And what's the intuition of that? 143 00:06:44,029 --> 00:06:46,069 Well, if we just look at it mathematically, if we have 0 144 00:06:46,069 --> 00:06:50,769 curl, somehow, the rotation in our x direction must be being 145 00:06:50,769 --> 00:06:53,229 offset by a rotation in the y direction. 146 00:06:53,230 --> 00:06:55,290 That the torques must be just perfectly 147 00:06:55,290 --> 00:06:56,210 offsetting each other. 148 00:06:56,209 --> 00:06:58,349 Let's see what happens if I were to change. 149 00:06:58,350 --> 00:06:59,740 So this was our old graph. 150 00:06:59,740 --> 00:07:03,000 Let me actually change it to my new vector field. 151 00:07:03,000 --> 00:07:06,310 152 00:07:06,310 --> 00:07:08,160 So that's our new vector field. 153 00:07:08,160 --> 00:07:12,700 This is our vector field plus yi plus xj. 154 00:07:12,699 --> 00:07:14,889 And now the curl is 0 everywhere. 155 00:07:14,889 --> 00:07:18,539 Which implies, or which means, that I could put a twig 156 00:07:18,540 --> 00:07:23,310 anywhere here and I'm not going to get any kind of rotation. 157 00:07:23,310 --> 00:07:24,160 Let's see if that makes sense. 158 00:07:24,160 --> 00:07:29,070 If I put a twig in the center, or some kind of stick in 159 00:07:29,069 --> 00:07:30,740 the center, let's see. 160 00:07:30,740 --> 00:07:34,329 I would have the forces or the fluids pushing in in this 161 00:07:34,329 --> 00:07:36,859 direction, but they're not helping to rotate, and pushing 162 00:07:36,860 --> 00:07:38,280 out in that direction. 163 00:07:38,279 --> 00:07:40,869 Well, that's not going to help me either. 164 00:07:40,870 --> 00:07:42,110 So I'm definitely not going to rotate there. 165 00:07:42,110 --> 00:07:44,069 And actually, you could put a twig anywhere. 166 00:07:44,069 --> 00:07:46,240 And maybe a twig might be pushed in a direction. 167 00:07:46,240 --> 00:07:48,519 For example, if I put a twig here, it's going to be pushed 168 00:07:48,519 --> 00:07:51,449 outward by the flow of the water, by the velocity 169 00:07:51,449 --> 00:07:52,349 of the water. 170 00:07:52,350 --> 00:07:53,920 But it's not going to rotate. 171 00:07:53,920 --> 00:07:54,720 So the [? ink ?] 172 00:07:54,720 --> 00:07:55,230 kind of holds. 173 00:07:55,230 --> 00:07:59,740 That's even though I kind of have curl in the x direction, 174 00:07:59,740 --> 00:08:01,300 or I have curl in the y direction, they're 175 00:08:01,300 --> 00:08:02,590 offsetting each other. 176 00:08:02,589 --> 00:08:04,719 So that in 2 dimensions, I actually end up 177 00:08:04,720 --> 00:08:06,990 having no rotation. 178 00:08:06,990 --> 00:08:10,069 And actually, this is called an irrotational-- I think that's 179 00:08:10,069 --> 00:08:11,409 the word-- vector field. 180 00:08:11,410 --> 00:08:13,510 Where you're not going to have any rotation here. 181 00:08:13,509 --> 00:08:17,519 All of the-- if you think of it as force or velocity of the 182 00:08:17,519 --> 00:08:21,349 vector field-- is going to be applying translation to 183 00:08:21,350 --> 00:08:23,870 objects in that field. 184 00:08:23,870 --> 00:08:25,519 And actually, just for fun, let's think about the 185 00:08:25,519 --> 00:08:26,629 divergence of that field. 186 00:08:26,629 --> 00:08:27,899 Just to, I don't know. 187 00:08:27,899 --> 00:08:30,250 Just because I have 2 minutes. 188 00:08:30,250 --> 00:08:33,179 So if I were to think about the divergence of that field, it's 189 00:08:33,179 --> 00:08:35,339 fairly easy to calculate. 190 00:08:35,340 --> 00:08:37,920 Let me erase this real fast. 191 00:08:37,919 --> 00:08:40,240 It's always fun to just interpret a vector 192 00:08:40,240 --> 00:08:41,399 field to death. 193 00:08:41,399 --> 00:08:44,319 Let's do the divergence of this one. 194 00:08:44,320 --> 00:08:47,000 So the divergence of that vector field, is just a 195 00:08:47,000 --> 00:08:50,210 partial derivative of this with respect to x. 196 00:08:50,210 --> 00:08:58,350 So div of v is the same thing as our del operator dot 197 00:08:58,350 --> 00:09:00,230 our vector field, v. 198 00:09:00,230 --> 00:09:01,810 And that's the partial derivative of this 199 00:09:01,809 --> 00:09:03,259 with respect to x. 200 00:09:03,259 --> 00:09:06,289 Well, the partial derivative of y with respect to x is just 0. 201 00:09:06,289 --> 00:09:11,139 Plus the partial derivative of x with respect to y. 202 00:09:11,139 --> 00:09:12,860 So this is 0. 203 00:09:12,860 --> 00:09:14,519 So the divergence of this field is 0. 204 00:09:14,519 --> 00:09:16,429 And even in the other case, when this was a minus, it 205 00:09:16,429 --> 00:09:18,239 still would have been 0. 206 00:09:18,240 --> 00:09:20,590 And does that make sense? 207 00:09:20,590 --> 00:09:23,110 Well, if we take a circle anywhere here-- let's take it 208 00:09:23,110 --> 00:09:25,050 in the center, because the center's the most interesting. 209 00:09:25,049 --> 00:09:26,299 Let's take a circle. 210 00:09:26,299 --> 00:09:29,689 So we do have some fluid or particles coming in at a 211 00:09:29,690 --> 00:09:32,940 certain velocity from the bottom right and the top left. 212 00:09:32,940 --> 00:09:35,410 But just as much is coming out through the top right 213 00:09:35,409 --> 00:09:38,100 and the bottom left. 214 00:09:38,100 --> 00:09:40,649 Whatever's coming in through here and here is leaving 215 00:09:40,649 --> 00:09:41,490 through there and there. 216 00:09:41,490 --> 00:09:46,330 So if I had an infinitesimally small circle, or sphere, in 217 00:09:46,330 --> 00:09:50,650 this vector field, I would have no net density increasing. 218 00:09:50,649 --> 00:09:53,500 Or nothing would be entering into that circle, or in any 219 00:09:53,500 --> 00:09:55,879 given amount of time the concentration of that 220 00:09:55,879 --> 00:09:57,059 circle wouldn't change. 221 00:09:57,059 --> 00:09:58,659 And that's true pretty much anywhere. 222 00:09:58,659 --> 00:10:03,059 If I were to draw a circle here, because the divergence 223 00:10:03,059 --> 00:10:05,629 is 0, it's telling us that whatever's coming in in 1 224 00:10:05,629 --> 00:10:08,059 direction is coming out in the other directions. 225 00:10:08,059 --> 00:10:11,539 So I'm not getting any denser, or any less dense. 226 00:10:11,539 --> 00:10:15,079 I have no divergence or convergence. 227 00:10:15,080 --> 00:10:16,040 So that's just interesting. 228 00:10:16,039 --> 00:10:17,120 Well, now I've run of time. 229 00:10:17,120 --> 00:10:19,149 So we're done analyzing this vector field. 230 00:10:19,149 --> 00:10:21,299 See you in the next video. 231 00:10:21,299 --> 00:10:21,899