1 00:00:00,000 --> 00:00:00,810 2 00:00:00,810 --> 00:00:01,820 Welcome back. 3 00:00:01,820 --> 00:00:03,690 Hopefully you have a little intuition now 4 00:00:03,690 --> 00:00:04,769 of what the curl is. 5 00:00:04,769 --> 00:00:08,070 Now let's actually compute it because if your sole goal is to 6 00:00:08,070 --> 00:00:11,990 pass a test and not understand the nature of the universe, 7 00:00:11,990 --> 00:00:14,820 which I think would be sad, but if that is your goal you at 8 00:00:14,820 --> 00:00:16,879 least need to know how to calculate these things. 9 00:00:16,879 --> 00:00:21,699 But it's even more fun when you have the intuition. 10 00:00:21,699 --> 00:00:23,419 And then you'll hopefully never forget it. 11 00:00:23,420 --> 00:00:26,480 We'll take the curl of a fairly fancy vector field. 12 00:00:26,480 --> 00:00:29,010 One that I have trouble visualizing but that we can 13 00:00:29,010 --> 00:00:31,020 mathematically chug through. 14 00:00:31,019 --> 00:00:35,460 So let's say our vector field-- and I'll do a three dimensional 15 00:00:35,460 --> 00:00:39,480 vector field just to do a fairly complicated example; I'm 16 00:00:39,479 --> 00:00:41,504 just going to make it up on the fly --so let's say in the 17 00:00:41,505 --> 00:00:45,545 x-direction the magnitude of the field is, I don't know, 18 00:00:45,545 --> 00:00:57,670 let's say it's x squared, y, sine, z, in the x-direction, 19 00:00:57,670 --> 00:01:06,790 plus-- I don't know --let's make it x, y squared, z in 20 00:01:06,790 --> 00:01:09,420 the j-direction, or the y-direction. 21 00:01:09,420 --> 00:01:13,409 And in the z-direction, I don't know, let's make it cosine of 22 00:01:13,409 --> 00:01:19,879 x times cosine of y, in the z-direction. 23 00:01:19,879 --> 00:01:25,459 Now we said that you can view the curl of this vector field-- 24 00:01:25,459 --> 00:01:27,640 and I have no intuition of what this vector field looks 25 00:01:27,640 --> 00:01:30,180 like; I just made this up. 26 00:01:30,180 --> 00:01:33,870 Maybe we'll graph it for fun just to see how messed up it 27 00:01:33,870 --> 00:01:38,439 looks --but we said this curl, you could view it as a cross 28 00:01:38,439 --> 00:01:45,230 product of our del operator and the vector field. 29 00:01:45,230 --> 00:01:47,990 Well, when you were using this engineering notation, when you 30 00:01:47,989 --> 00:01:53,619 have a vector broken down into it x, y, and z components, or 31 00:01:53,620 --> 00:01:57,130 it's i, j, and k components, you can take the determinant of 32 00:01:57,129 --> 00:01:59,489 that matrix-- when I showed you how to compute the cross 33 00:01:59,489 --> 00:02:01,619 product --to figure out the cross product. 34 00:02:01,620 --> 00:02:02,590 So how do we do this? 35 00:02:02,590 --> 00:02:11,439 So the cross product is going to be equal to-- I didn't have 36 00:02:11,439 --> 00:02:15,969 to draw a straight line --so how you take the cross product 37 00:02:15,969 --> 00:02:19,680 of this vector field and the gradient operator? 38 00:02:19,680 --> 00:02:23,990 Well, you write i, j, k on top like you're taking the cross 39 00:02:23,990 --> 00:02:29,680 product of any two three dimensional vectors, and then 40 00:02:29,680 --> 00:02:34,099 you take the first vector-- but it's really a vector operator, 41 00:02:34,099 --> 00:02:38,409 but it's this del operator. 42 00:02:38,409 --> 00:02:40,250 And what are the components of the del operator? 43 00:02:40,250 --> 00:02:44,270 It's the partial derivative with respect to x, the partial 44 00:02:44,270 --> 00:02:48,600 derivative with respect to y, partial derivative with 45 00:02:48,599 --> 00:02:50,159 respect to z, right? 46 00:02:50,159 --> 00:02:52,750 Let me just rewrite the del operator. 47 00:02:52,750 --> 00:02:55,349 You can view it as being equal to the partial with respect to 48 00:02:55,349 --> 00:03:01,750 x, i plus the partial with respect to y, j plus the 49 00:03:01,750 --> 00:03:05,629 partial with respect to z, k. 50 00:03:05,629 --> 00:03:07,900 So its x, y, and z components are the partials with respect 51 00:03:07,900 --> 00:03:10,390 to x, with respect to y, with respect to z. 52 00:03:10,389 --> 00:03:13,159 And then the second, where we're taking this operator 53 00:03:13,159 --> 00:03:14,375 cross the vector field. 54 00:03:14,375 --> 00:03:16,500 So what are the components of the vector field? 55 00:03:16,500 --> 00:03:25,129 I'll probably run out of space, but it's x squared, y, sine, z. 56 00:03:25,129 --> 00:03:29,939 Then here it's xy squared, z-- I should written all of this 57 00:03:29,939 --> 00:03:33,039 bigger --and then the third column, the z component is 58 00:03:33,039 --> 00:03:36,979 cosine of x times cosine of y. 59 00:03:36,979 --> 00:03:39,799 Just the x, y, and z components. 60 00:03:39,800 --> 00:03:43,350 And now we are ready to take the determinant, which will 61 00:03:43,349 --> 00:03:46,259 probably, well, it'll probably get pretty 62 00:03:46,259 --> 00:03:47,989 messy, but let's try it. 63 00:03:47,990 --> 00:03:55,710 So this is equal to the i-unit vector-- let me use a more 64 00:03:55,710 --> 00:03:59,245 vibrant color --the i-unit vector times it's 65 00:03:59,245 --> 00:03:59,500 subdeterminant. 66 00:03:59,500 --> 00:04:02,740 So you cross out it's row and column, and so you take the 67 00:04:02,740 --> 00:04:06,540 determinant of this expression, so it's going to be times-- 68 00:04:06,539 --> 00:04:09,129 this times this, but it's really the partial. 69 00:04:09,129 --> 00:04:12,400 If you multiply the partial with respect to y operator 70 00:04:12,400 --> 00:04:14,986 times that expression, you're really just taking, since it's 71 00:04:14,985 --> 00:04:17,240 an operator and not an expression, you're really just 72 00:04:17,240 --> 00:04:19,150 going to take the partial of this with respect to y, 73 00:04:19,149 --> 00:04:19,969 but I'll write it down. 74 00:04:19,970 --> 00:04:26,750 So it's going to be the partial with respect to y of cosine x, 75 00:04:26,750 --> 00:04:37,019 cosine y minus the partial with respect to z times xy squared, 76 00:04:37,019 --> 00:04:45,490 z-- and now we're on to our j component --plus j. 77 00:04:45,490 --> 00:04:48,800 So now what's the magnitude of our curl in the j-direction? 78 00:04:48,800 --> 00:04:50,860 Let's cross out the row in the column of j. 79 00:04:50,860 --> 00:04:54,580 So it's a partial with respect to x of this. 80 00:04:54,579 --> 00:04:58,129 So this is maybe messier than I originally intended. 81 00:04:58,129 --> 00:05:05,699 Cosine of x, cosine of y, cross these columns there, minus 82 00:05:05,699 --> 00:05:20,539 the partial with respect to z of x squared, y sine of z. 83 00:05:20,540 --> 00:05:23,370 And then finally our k component. 84 00:05:23,370 --> 00:05:28,100 Oh, and sorry, when you take the determinant, you use that-- 85 00:05:28,100 --> 00:05:31,180 and this is all kind of a bit of voodoo --but you put a plus 86 00:05:31,180 --> 00:05:33,550 here, a minus here, a plus here, so it's kind of 87 00:05:33,550 --> 00:05:34,350 this checkered pattern. 88 00:05:34,350 --> 00:05:37,629 To this is plus i, this should actually be minus j. 89 00:05:37,629 --> 00:05:40,420 Don't want to make that mistake; this is minus j. 90 00:05:40,420 --> 00:05:46,340 This is just kind of the algorithm of how do you 91 00:05:46,339 --> 00:05:48,179 take a determinant. 92 00:05:48,180 --> 00:05:53,730 OK, then finally we have plus k times the determinant 93 00:05:53,730 --> 00:05:57,110 of its submatrix. 94 00:05:57,110 --> 00:06:06,660 So the partial with respect to x of this, sorry, we take out 95 00:06:06,660 --> 00:06:13,040 it's row and column, so xy squared, z minus the partial 96 00:06:13,040 --> 00:06:19,189 with respect to y of this. 97 00:06:19,189 --> 00:06:20,990 Why don't we take this row and columns, and this is the 98 00:06:20,990 --> 00:06:28,780 submatrix of x squared, y, sine of z. 99 00:06:28,779 --> 00:06:30,869 All right, now let me try to simplify it, and I'll 100 00:06:30,870 --> 00:06:31,860 have to get some space. 101 00:06:31,860 --> 00:06:34,960 Hopefully you understood what I did here and now we got this. 102 00:06:34,959 --> 00:06:36,930 And now I think I can erase all of this and just so I 103 00:06:36,930 --> 00:06:39,840 can have some room to simplify things in. 104 00:06:39,839 --> 00:06:43,089 105 00:06:43,089 --> 00:06:46,539 No, that's not what I wanted to do; I wanted to do 106 00:06:46,540 --> 00:06:49,569 it in a darker color. 107 00:06:49,569 --> 00:06:51,909 That's what I wanted to do. 108 00:06:51,910 --> 00:06:54,320 Erase that. 109 00:06:54,319 --> 00:06:55,029 Erase that. 110 00:06:55,029 --> 00:06:57,750 Now we just have to simplify it, taking a bunch of 111 00:06:57,750 --> 00:06:58,259 partial derivatives. 112 00:06:58,259 --> 00:07:02,629 What's the partial derivative of this with respect to y? 113 00:07:02,629 --> 00:07:08,319 Well, x is just a constant, so it's going to be-- well, we can 114 00:07:08,319 --> 00:07:10,480 just put the i out front, but eventually we want to write our 115 00:07:10,480 --> 00:07:14,350 magnitude before the vector --so it's i times the partial 116 00:07:14,350 --> 00:07:17,720 of this with respect to y with our constant. 117 00:07:17,720 --> 00:07:19,850 Cosine of x is just a constant. 118 00:07:19,850 --> 00:07:21,800 And then what's the derivative of this with respect to y? 119 00:07:21,800 --> 00:07:24,090 It's minus sine of y. 120 00:07:24,089 --> 00:07:30,549 So we'll write sine of y, and let's put the minus out front; 121 00:07:30,550 --> 00:07:32,170 these are what's multiplied. 122 00:07:32,170 --> 00:07:39,310 OK, and then we have minus, now we stick the partial with 123 00:07:39,310 --> 00:07:42,180 respect-- sorry, actually, I forgot to do this part. 124 00:07:42,180 --> 00:07:46,509 Let me start over, actually. 125 00:07:46,509 --> 00:07:49,050 So let me just take this expression and I'll 126 00:07:49,050 --> 00:07:49,634 multiply it by i. 127 00:07:49,634 --> 00:07:55,180 The partial of this with respect to y is cosine of x 128 00:07:55,180 --> 00:07:58,590 times minus sine of y-- let's put the minus out front 129 00:07:58,589 --> 00:07:59,759 --minus sine of y. 130 00:07:59,759 --> 00:08:02,289 131 00:08:02,290 --> 00:08:05,080 Now minus the partial of this with respect to z. 132 00:08:05,079 --> 00:08:08,779 Well the partial of this with respect to z, xy squared 133 00:08:08,779 --> 00:08:10,109 is just a constant, right? 134 00:08:10,110 --> 00:08:11,560 So the partial of this with respect to z 135 00:08:11,560 --> 00:08:13,530 is just xy squared. 136 00:08:13,529 --> 00:08:18,359 So minus xy squared, and then we're going to have all of 137 00:08:18,360 --> 00:08:23,270 that; that's the magnitude in our i-direction. 138 00:08:23,269 --> 00:08:27,169 And now we have minus-- because minus in the j-direction 139 00:08:27,170 --> 00:08:30,490 --what's the partial derivative of this with respect to x? 140 00:08:30,490 --> 00:08:32,483 Well, the partial of cosine of x with respect to 141 00:08:32,482 --> 00:08:33,778 x is minus sine of x. 142 00:08:33,778 --> 00:08:37,470 143 00:08:37,470 --> 00:08:39,970 And cosine of y is just a constant, so it just 144 00:08:39,970 --> 00:08:43,080 carries over, cosine of y. 145 00:08:43,080 --> 00:08:47,240 And then that should be-- oh, there we go --minus this 146 00:08:47,240 --> 00:08:50,350 expression, the partial of this with respect to z. 147 00:08:50,350 --> 00:08:52,779 Well, the derivative of sine of z with respect 148 00:08:52,779 --> 00:08:53,959 to z is cosine of z. 149 00:08:53,960 --> 00:08:57,660 This is just a constant, so it's minus x squared, 150 00:08:57,659 --> 00:09:00,669 y, cosine of z. 151 00:09:00,669 --> 00:09:03,159 And that's the magnitude in the j-direction. 152 00:09:03,159 --> 00:09:04,500 We're almost there. 153 00:09:04,500 --> 00:09:08,029 And now finally, plus, what's the partial of 154 00:09:08,029 --> 00:09:09,990 this with respect to x? 155 00:09:09,990 --> 00:09:17,480 Well, these are just constants, so it's y squared, z minus-- 156 00:09:17,480 --> 00:09:19,600 once again, we just have a y term; everything else is a 157 00:09:19,600 --> 00:09:22,680 constant --so the partial with respect to y is 158 00:09:22,679 --> 00:09:27,250 x squared, sine of z. 159 00:09:27,250 --> 00:09:31,961 And that's the magnitude in our k-direction. 160 00:09:31,961 --> 00:09:33,299 And we're pretty much done. 161 00:09:33,299 --> 00:09:36,569 I mean we can simplify it a little bit just 162 00:09:36,570 --> 00:09:38,280 to make it clean. 163 00:09:38,279 --> 00:09:40,419 And essentially we could just, I don't have to rewrite it, 164 00:09:40,419 --> 00:09:43,129 we can just multiply this by negative 1. 165 00:09:43,129 --> 00:09:46,939 So then this becomes plus, plus, plus. 166 00:09:46,940 --> 00:09:48,310 Yeah, that's pretty much it. 167 00:09:48,309 --> 00:09:52,049 This is the curl of the vector field v at any 168 00:09:52,049 --> 00:09:53,449 point x, y, and z. 169 00:09:53,450 --> 00:09:54,590 So that's how you calculate it. 170 00:09:54,590 --> 00:09:59,810 You just literally take the cross product of that del 171 00:09:59,809 --> 00:10:02,019 operator and your vector field. 172 00:10:02,019 --> 00:10:04,350 And you'll get something fairly hairy, although this was, I 173 00:10:04,350 --> 00:10:06,700 think, a hairier than average problem. 174 00:10:06,700 --> 00:10:09,320 In the next video, we'll do a little bit of this, but I think 175 00:10:09,320 --> 00:10:13,120 it'll give you more intuition and less of just the algorithm 176 00:10:13,120 --> 00:10:15,019 and the computation of how do you do it. 177 00:10:15,019 --> 00:10:17,100 So you in the next video.