1 00:00:00,000 --> 00:00:00,460 2 00:00:00,460 --> 00:00:03,540 What I want to do in this video is establish a reasonably 3 00:00:03,540 --> 00:00:07,540 powerful condition in which we can establish that at vector 4 00:00:07,540 --> 00:00:10,349 field, or that a line integral of a vector field is 5 00:00:10,349 --> 00:00:11,669 path independent. 6 00:00:11,669 --> 00:00:14,419 And when I say that, I mean that let's say I were to take 7 00:00:14,419 --> 00:00:21,780 this line integral along the path c of f dot d r, and let's 8 00:00:21,780 --> 00:00:24,310 say my path looks like this. 9 00:00:24,309 --> 00:00:28,109 10 00:00:28,109 --> 00:00:34,019 That's my x and y axis, and let's say my path looks 11 00:00:34,020 --> 00:00:37,530 something like this: I start there and I go over 12 00:00:37,530 --> 00:00:40,509 there to point c. 13 00:00:40,509 --> 00:00:43,049 My end point, the curve here is c. 14 00:00:43,049 --> 00:00:46,209 And so I would evaluate this line integral, this victor 15 00:00:46,210 --> 00:00:48,640 field along this path. 16 00:00:48,640 --> 00:00:52,420 This would be a path independent vector field, or we 17 00:00:52,420 --> 00:00:59,890 call that a conservative vector field, if this thing is equal 18 00:00:59,890 --> 00:01:06,370 to the same integral over a different path that has 19 00:01:06,370 --> 00:01:07,240 the same end point. 20 00:01:07,239 --> 00:01:11,250 So let's call this c1, so this is c1, and this is c2. 21 00:01:11,250 --> 00:01:15,680 This vector field is conservative if I start 22 00:01:15,680 --> 00:01:17,530 at the same point but I take a different path. 23 00:01:17,530 --> 00:01:22,049 Let's say I go something like that: if I take a different 24 00:01:22,049 --> 00:01:27,599 path-- and this is my c2 --I still get the same value. 25 00:01:27,599 --> 00:01:30,189 What this is telling me is that all it cares about to evaluate 26 00:01:30,189 --> 00:01:33,980 these integrals is my starting point and my ending point. 27 00:01:33,980 --> 00:01:35,750 It doesn't care what I do in between. 28 00:01:35,750 --> 00:01:38,409 It doesn't care how I get from my starting point 29 00:01:38,409 --> 00:01:40,140 to my end point. 30 00:01:40,140 --> 00:01:42,609 These two integrals have the same start point and same end 31 00:01:42,609 --> 00:01:45,319 point, so irregardless of their actual path, they're 32 00:01:45,319 --> 00:01:46,349 going to be the same. 33 00:01:46,349 --> 00:01:50,530 That's what it means for f to be a conservative field, or 34 00:01:50,530 --> 00:01:54,670 what it means for this integral to be path independent. 35 00:01:54,670 --> 00:01:58,079 So before I prove or I show you the conditions, let's build 36 00:01:58,079 --> 00:02:00,560 up our tool kit a little bit. 37 00:02:00,560 --> 00:02:04,750 And so you may or may not have already seen the 38 00:02:04,750 --> 00:02:06,590 multivariable chain rule. 39 00:02:06,590 --> 00:02:11,219 40 00:02:11,219 --> 00:02:14,849 And I'm not going to prove it in this video, but I 41 00:02:14,849 --> 00:02:16,389 think it'll be pretty intuitive for you. 42 00:02:16,389 --> 00:02:19,000 So maybe it doesn't need to have a proof, or I'll prove it 43 00:02:19,000 --> 00:02:21,560 eventually, but I really just want to give you the intuition. 44 00:02:21,560 --> 00:02:24,610 And all that says is that if I have some function-- let's say 45 00:02:24,610 --> 00:02:29,860 I have f of x and y, but x and y are then functions of, let's 46 00:02:29,860 --> 00:02:37,220 say a third variable, t, so f of x of t and y of t --that the 47 00:02:37,219 --> 00:02:45,719 derivative of f with respect to t is multivariable. 48 00:02:45,719 --> 00:02:48,000 I have two variables here in x and y. 49 00:02:48,000 --> 00:02:50,810 This is going to be equal to the partial of f 50 00:02:50,810 --> 00:02:52,349 with respect to x. 51 00:02:52,349 --> 00:02:58,129 How fast does f change as x changes times the derivative of 52 00:02:58,129 --> 00:03:02,120 x with respect to t-- This is a single variable function right 53 00:03:02,120 --> 00:03:04,780 here, so you to take a regular derivative. 54 00:03:04,780 --> 00:03:09,140 So times how fast x changes with respect to t. 55 00:03:09,139 --> 00:03:11,679 This is a standard derivative, this is a partial derivative, 56 00:03:11,680 --> 00:03:14,290 because at that level we're dealing with two variables. 57 00:03:14,289 --> 00:03:18,280 And we're not done. --plus how fast f changes with respect to 58 00:03:18,280 --> 00:03:25,960 y times the derivative y with respected t. 59 00:03:25,960 --> 00:03:29,110 So d y d t. 60 00:03:29,110 --> 00:03:31,490 And I'm not going to prove it, but I think it makes 61 00:03:31,490 --> 00:03:32,730 pretty good intuition. 62 00:03:32,729 --> 00:03:35,949 This is saying, as I move a little bit d t, how 63 00:03:35,949 --> 00:03:37,289 much of a d f do I get? 64 00:03:37,289 --> 00:03:40,199 Or how fast this f change with respect to t? 65 00:03:40,199 --> 00:03:42,399 It says, well there's two ways that f can change: it can 66 00:03:42,400 --> 00:03:45,670 change with respect to x and it can change with respect to y. 67 00:03:45,669 --> 00:03:49,169 So why don't I add those two things together as they're both 68 00:03:49,169 --> 00:03:51,019 changing with respect to t? 69 00:03:51,020 --> 00:03:53,939 That's all it's saying, and if you kind of imagined that you 70 00:03:53,939 --> 00:03:57,090 could cancel out this partial x with this d x, and this partial 71 00:03:57,090 --> 00:04:00,550 y with this d y, you could kind of imagine the partial of f 72 00:04:00,550 --> 00:04:04,200 with respect to t on the x side of things, and then plus the 73 00:04:04,199 --> 00:04:08,649 partial of f with respect to t in the y dimension. 74 00:04:08,650 --> 00:04:10,930 And then that'll give you the total change of f 75 00:04:10,930 --> 00:04:12,150 with respected to t. 76 00:04:12,150 --> 00:04:15,560 Kind of a hand-wavy argument there, but at least to me this 77 00:04:15,560 --> 00:04:16,899 is a pretty intuitive formula. 78 00:04:16,899 --> 00:04:19,419 So that's our tool kit right there; the multivariable 79 00:04:19,420 --> 00:04:20,449 chain rule. 80 00:04:20,449 --> 00:04:23,039 We're going to put that aside for a second. 81 00:04:23,040 --> 00:04:29,090 Now let's say I have some vector field f-- and it's 82 00:04:29,089 --> 00:04:31,099 different than this f, so I'll do it in a different color; 83 00:04:31,100 --> 00:04:34,660 magenta --I have some vector field f that is a 84 00:04:34,660 --> 00:04:36,920 function of x and y. 85 00:04:36,920 --> 00:04:40,379 And let's say that it happens to be the gradient of 86 00:04:40,379 --> 00:04:41,704 some scalar field. 87 00:04:41,704 --> 00:04:46,909 88 00:04:46,910 --> 00:04:49,439 I'll call that capital F. 89 00:04:49,439 --> 00:04:53,060 And this is gradient which means that capital F is also 90 00:04:53,060 --> 00:04:59,230 function of x and y-- so I don't want to write it on a new 91 00:04:59,230 --> 00:05:02,890 line, I could also write up here; capital F is also a 92 00:05:02,889 --> 00:05:07,469 function of x and y --and the gradient, and all that means is 93 00:05:07,470 --> 00:05:13,150 that the vector field f of xy-- lower-case f of xy, is equal to 94 00:05:13,149 --> 00:05:18,709 the partial derivative of upper-case F with respect to x 95 00:05:18,709 --> 00:05:24,250 times the i-unit vector plus the partial of upper-case F 96 00:05:24,250 --> 00:05:27,689 with respect to y times the j-unit vector. 97 00:05:27,689 --> 00:05:32,170 This is the definition of the gradient right here. 98 00:05:32,170 --> 00:05:35,480 And if you imagine that upper-case F is some type of 99 00:05:35,480 --> 00:05:48,530 surface-- so this is uppercase F of xy --the gradient F of xy 100 00:05:48,529 --> 00:05:51,389 is going to be a vector field that tells you the direction of 101 00:05:51,389 --> 00:05:53,439 steepest descent at any point. 102 00:05:53,439 --> 00:05:56,740 So it'll be defined the xy plane. 103 00:05:56,740 --> 00:06:00,060 So on the xy plane it'll tell you-- so let me draw; that's 104 00:06:00,060 --> 00:06:04,819 the vertical axis, maybe that's the x axis, that's the y axis 105 00:06:04,819 --> 00:06:08,149 --so the gradient of it, if you take any point on the xy plane, 106 00:06:08,149 --> 00:06:10,239 it'll tell you the direction you need to travel to go 107 00:06:10,240 --> 00:06:11,610 into the deepest descent. 108 00:06:11,610 --> 00:06:13,319 And for this gradient field it's going to be 109 00:06:13,319 --> 00:06:14,699 something like this. 110 00:06:14,699 --> 00:06:18,449 111 00:06:18,449 --> 00:06:20,729 And maybe over here it starts going in that direction because 112 00:06:20,730 --> 00:06:23,800 you would descend towards this little minimum 113 00:06:23,800 --> 00:06:24,259 point right here. 114 00:06:24,259 --> 00:06:25,769 Anyway, I don't want to get too involved. 115 00:06:25,769 --> 00:06:28,959 And the whole point of this isn't to really get the 116 00:06:28,959 --> 00:06:32,049 intuition behind gradients; there are other videos on this. 117 00:06:32,050 --> 00:06:35,960 The point of this is to get other a test to see whether 118 00:06:35,959 --> 00:06:39,289 something is path independent; whether a vector field is path 119 00:06:39,290 --> 00:06:41,050 independent, whether it's conservative. 120 00:06:41,050 --> 00:06:45,009 And it turns out that if this exists-- and I'm going to prove 121 00:06:45,009 --> 00:06:57,349 it now --if f is the gradient of some scalar field, 122 00:06:57,350 --> 00:07:01,629 then f is conservative. 123 00:07:01,629 --> 00:07:04,569 124 00:07:04,569 --> 00:07:07,899 Or you could say it doesn't matter what path we follow when 125 00:07:07,899 --> 00:07:11,669 we take a line integral over f, it just matters about our 126 00:07:11,670 --> 00:07:14,949 starting point and our ending point. 127 00:07:14,949 --> 00:07:18,029 Now let me see if I can prove that to you. 128 00:07:18,029 --> 00:07:20,819 So let's start with the assumption that f can be 129 00:07:20,819 --> 00:07:24,560 written this way, as the gradient, that lower-case f can 130 00:07:24,560 --> 00:07:28,120 be written as the gradient of some upper-case F. 131 00:07:28,120 --> 00:07:36,490 So in that case our integral-- well, let's define 132 00:07:36,490 --> 00:07:37,509 our path first. 133 00:07:37,509 --> 00:07:41,310 So our position vector function-- we always need one 134 00:07:41,310 --> 00:07:45,250 of those to do a line integral or a vector line integral --r 135 00:07:45,250 --> 00:07:52,889 of t is going to be equal to x of t times i plus y of t times 136 00:07:52,889 --> 00:07:56,769 j 4t going between a and b. 137 00:07:56,769 --> 00:07:59,729 You've seen this multiple times; this is a definition 138 00:07:59,730 --> 00:08:04,950 of pretty much any path in two dimensions. 139 00:08:04,949 --> 00:08:13,269 And then we're going to say f of xy is going to be equal to 140 00:08:13,269 --> 00:08:16,759 this: it's going to be the partial derivative of uppercase 141 00:08:16,759 --> 00:08:20,740 F with respect to x-- so we're assuming that this exists, that 142 00:08:20,740 --> 00:08:25,790 this is true --times i plus the partial of upper-case F 143 00:08:25,790 --> 00:08:29,210 with respect to y times j. 144 00:08:29,209 --> 00:08:42,470 Now, given this what is lower-case f dot dr going 145 00:08:42,470 --> 00:08:45,080 to equal over this path right here? 146 00:08:45,080 --> 00:08:48,660 This path is defined by this position function right there. 147 00:08:48,659 --> 00:08:51,969 Well, it's going to be equal to, we need to figure out 148 00:08:51,970 --> 00:08:54,279 what dr is, and we've done that in multiple videos. 149 00:08:54,279 --> 00:08:56,480 I'll do that on the right over here. 150 00:08:56,480 --> 00:08:59,740 dr, we've seen it multiple times. 151 00:08:59,740 --> 00:09:00,980 Actually, I'll solve it out again. 152 00:09:00,980 --> 00:09:11,590 dr over dt by definition was equal to dx over dt times i 153 00:09:11,590 --> 00:09:15,960 plus-- I don't know why it got all fat like that 154 00:09:15,960 --> 00:09:20,070 --dy over dt times j. 155 00:09:20,070 --> 00:09:21,629 That's what dr over dt is. 156 00:09:21,629 --> 00:09:24,320 So if we want to figure out what dr is, the differential of 157 00:09:24,320 --> 00:09:26,720 dr, if we want to play with differentials in this way, 158 00:09:26,720 --> 00:09:29,490 multiply both sides times dt. 159 00:09:29,490 --> 00:09:31,519 And actually I'm going to treat dt, I'll multiply 160 00:09:31,519 --> 00:09:32,669 it; I'll distribute it. 161 00:09:32,669 --> 00:09:44,029 It's dx over dt times dti plus dy over dt times dtj. 162 00:09:44,029 --> 00:09:48,459 So if we're taking the dot product of f with dr, what 163 00:09:48,460 --> 00:09:49,060 are we going to get? 164 00:09:49,059 --> 00:09:55,079 165 00:09:55,080 --> 00:10:01,960 So this is going to be the integral over the curve from-- 166 00:10:01,960 --> 00:10:04,840 I'll write the c right there; we could write in terms of the 167 00:10:04,840 --> 00:10:08,790 end points as t once we feel good that we have everything in 168 00:10:08,789 --> 00:10:12,559 terms of t --but it's going to be equal to this dot that, 169 00:10:12,559 --> 00:10:16,509 which is equal to-- I'll try to stay color consistent --the 170 00:10:16,509 --> 00:10:23,450 partial of upper-case F with respect to x times that, times 171 00:10:23,450 --> 00:10:26,520 dx over d t-- I'm going to write this st in a different 172 00:10:26,519 --> 00:10:35,850 color --times dt plus the partial of upper-case F with 173 00:10:35,850 --> 00:10:41,019 respect to y times-- we're multiplying the 174 00:10:41,019 --> 00:10:41,949 j components, right? 175 00:10:41,950 --> 00:10:45,170 When you take the dot product, multiply the i components, and 176 00:10:45,169 --> 00:10:48,679 then add that to what you get from the product of the j 177 00:10:48,679 --> 00:10:50,399 components --so this j component's partial of 178 00:10:50,399 --> 00:10:54,829 upper-case F with respect to y, and then we have times-- switch 179 00:10:54,830 --> 00:11:03,680 to a yellow --dy over dt times that dt right over there. 180 00:11:03,679 --> 00:11:05,789 And then we can factor out the dt. 181 00:11:05,789 --> 00:11:08,860 182 00:11:08,860 --> 00:11:11,300 Or actually, so I don't have to even write it again, right now 183 00:11:11,299 --> 00:11:12,799 I wrote it without, well let me write it again. 184 00:11:12,799 --> 00:11:16,034 So this is equal to the integral. 185 00:11:16,034 --> 00:11:18,419 And let's say we have it in terms of t; we've written 186 00:11:18,419 --> 00:11:22,759 everything in terms of t, so t goes from a to b, and so this 187 00:11:22,759 --> 00:11:27,149 is going to be equal to-- I'll write it in blue --the partial 188 00:11:27,149 --> 00:11:33,610 of upper case F with respect to x times dx over dt plus-- I'm 189 00:11:33,610 --> 00:11:36,980 distributing this dt out --plus the partial of uppercase 190 00:11:36,980 --> 00:11:39,909 F with respect to y. 191 00:11:39,909 --> 00:11:42,199 dy over dt. 192 00:11:42,200 --> 00:11:46,530 all of that times dt. 193 00:11:46,529 --> 00:11:49,189 This is equivalent to that. 194 00:11:49,190 --> 00:11:51,950 Now you might realize why I talked about the 195 00:11:51,950 --> 00:11:53,640 multivariable chain rule. 196 00:11:53,639 --> 00:11:55,480 What is this right here? 197 00:11:55,480 --> 00:11:57,100 What is that right there? 198 00:11:57,100 --> 00:11:59,050 You can do some pattern matching. 199 00:11:59,049 --> 00:12:04,179 That is the same thing as the derivative of upper-case 200 00:12:04,179 --> 00:12:05,949 F with respect to t. 201 00:12:05,950 --> 00:12:07,980 Look at this; let me let me copy and paste that just 202 00:12:07,980 --> 00:12:08,940 so you appreciate it. 203 00:12:08,940 --> 00:12:17,170 204 00:12:17,169 --> 00:12:21,370 So this is our definition, or this is our-- I won't say 205 00:12:21,370 --> 00:12:22,919 definition; one can actually prove it. 206 00:12:22,919 --> 00:12:25,039 You don't have to start from there ---but this is our 207 00:12:25,039 --> 00:12:27,219 multivariable chain rule right here. 208 00:12:27,220 --> 00:12:29,889 The driven of any function with respect t is the partial of 209 00:12:29,889 --> 00:12:32,399 that function with respect to x times dx over dt plus the 210 00:12:32,399 --> 00:12:34,829 partial of that function with respect to dy over dt. 211 00:12:34,830 --> 00:12:37,710 I have the partial of upper-case F with respect to 212 00:12:37,710 --> 00:12:40,650 x times dx over dt plus the partial upper-case F 213 00:12:40,649 --> 00:12:42,129 with respect to y. 214 00:12:42,129 --> 00:12:44,340 This and this are identical if you just replace this 215 00:12:44,340 --> 00:12:46,420 lower-case f with an upper-case F. 216 00:12:46,419 --> 00:12:49,469 So this in blue right here, this whole expression is equal 217 00:12:49,470 --> 00:12:57,500 to the integral from t is equal to a to t is equal to b of-- in 218 00:12:57,500 --> 00:13:07,250 blue here --the derivative of f with respect to dt. 219 00:13:07,250 --> 00:13:10,029 And how do you evaluate-- let me just the dt in green 220 00:13:10,029 --> 00:13:11,980 --how do you evaluate something like this? 221 00:13:11,980 --> 00:13:14,789 I just want to make a point: this is just this from the 222 00:13:14,789 --> 00:13:16,539 multivariable chain rule. 223 00:13:16,539 --> 00:13:19,610 And how do you evaluate a definite integral like this? 224 00:13:19,610 --> 00:13:21,430 Well, you take the antiderivative of the 225 00:13:21,429 --> 00:13:25,120 inside with respect to dt. 226 00:13:25,120 --> 00:13:26,730 So what is this going to be equal to? 227 00:13:26,730 --> 00:13:32,029 You take the antiderivative of the inside, that's just f. 228 00:13:32,029 --> 00:13:36,192 So this is equal to f of t. 229 00:13:36,192 --> 00:13:38,409 And let me be clear. 230 00:13:38,409 --> 00:13:41,689 We wrote before that f is a function. 231 00:13:41,690 --> 00:13:47,330 So our upper-case F is a function of x and y, which 232 00:13:47,330 --> 00:13:49,389 could also be written, since each of these are functions 233 00:13:49,389 --> 00:13:54,309 of t, could be written as f of x of t of y of t. 234 00:13:54,309 --> 00:13:55,919 I'm just rewriting it in different ways. 235 00:13:55,919 --> 00:14:00,429 And this could be just written as f or t. 236 00:14:00,429 --> 00:14:02,549 These are all equivalent, depending on whether you want 237 00:14:02,549 --> 00:14:04,490 to include the x's or the y's only, or the t's 238 00:14:04,490 --> 00:14:06,310 only, or them both. 239 00:14:06,309 --> 00:14:10,369 Because both of the x's and y's are functions of t. 240 00:14:10,370 --> 00:14:13,649 So this is the derivative of f with respect to t. 241 00:14:13,649 --> 00:14:15,879 If this was just in terms of t, this is the derivative 242 00:14:15,879 --> 00:14:17,389 of that with respect to t. 243 00:14:17,389 --> 00:14:20,879 We take its antiderivative, we're left just with f, and we 244 00:14:20,879 --> 00:14:26,539 have to evaluate it from t is equal to a to t is equal to b. 245 00:14:26,539 --> 00:14:29,730 And so this is equal to-- and this is the home stretch 246 00:14:29,730 --> 00:14:35,909 --this is equal to f of b minus f of a. 247 00:14:35,909 --> 00:14:38,169 And if you want to think about it in these terms, this 248 00:14:38,169 --> 00:14:39,259 is the same thing. 249 00:14:39,259 --> 00:14:46,559 This is equal to f of x of b over y of b-- let me make sure 250 00:14:46,559 --> 00:14:49,969 I got all the parentheses --minus f of x of 251 00:14:49,970 --> 00:14:52,330 a over y of a. 252 00:14:52,330 --> 00:14:53,660 These are equivalent. 253 00:14:53,659 --> 00:14:58,319 You give me any point on the xy plane, an x and a y, and 254 00:14:58,320 --> 00:15:00,070 it tells me where I am. 255 00:15:00,070 --> 00:15:03,220 This is my capital F, it gives me a height. 256 00:15:03,220 --> 00:15:05,800 Just like that. 257 00:15:05,799 --> 00:15:09,651 This associates a value with every point on the xy plane. 258 00:15:09,652 --> 00:15:13,370 But this whole exercise, remember this is the 259 00:15:13,370 --> 00:15:15,279 same thing as that. 260 00:15:15,279 --> 00:15:18,009 This is our whole thing that we were trying to prove: that is 261 00:15:18,009 --> 00:15:25,299 equal to f dot dr. f dot dr, our vector field, which is the 262 00:15:25,299 --> 00:15:29,889 gradient of the capital F-- remember F was equal to the 263 00:15:29,889 --> 00:15:33,000 gradient of F, we assume that it's the gradient of some 264 00:15:33,000 --> 00:15:36,799 function capital F, if that is the case, then we just did a 265 00:15:36,799 --> 00:15:39,179 little bit of calculus or algebra, whatever you want to 266 00:15:39,179 --> 00:15:43,149 call it, and we found that we can evaluate this integral by 267 00:15:43,149 --> 00:15:47,169 evaluating capital F at t is equal to b, and then 268 00:15:47,169 --> 00:15:50,569 subtracting from that capital F at t is equal to a. 269 00:15:50,570 --> 00:15:52,860 But what that tells you is that this integral, the value of 270 00:15:52,860 --> 00:15:58,980 this integral, is only dependent at our starting 271 00:15:58,980 --> 00:16:06,090 point, t is equal to a, this is the point x of a, y of a, and 272 00:16:06,090 --> 00:16:13,810 the ending point, t is equal to b, which is x of b, y of b. 273 00:16:13,809 --> 00:16:16,019 That integral is only dependent on these two values. 274 00:16:16,019 --> 00:16:17,299 How do I know that? 275 00:16:17,299 --> 00:16:20,339 Because to solve it-- because I'm saying that this thing 276 00:16:20,340 --> 00:16:23,879 exists --I just had to evaluate that thing at those two 277 00:16:23,879 --> 00:16:26,889 points; I didn't care about the curve in between. 278 00:16:26,889 --> 00:16:34,449 So this shows that if F is equal to the gradient-- this 279 00:16:34,450 --> 00:16:37,340 is often called a potential function of capital F, although 280 00:16:37,340 --> 00:16:39,590 they're usually the negative each other, but it's the same 281 00:16:39,590 --> 00:16:44,220 idea --if the vector field f is the gradient of some scale or 282 00:16:44,220 --> 00:16:56,490 field upper-case F, then we can say that f is conservative or 283 00:16:56,490 --> 00:17:00,980 that the line integral of f dot dr is path independent. 284 00:17:00,980 --> 00:17:04,420 It doesn't matter what path we go on as long as our starting 285 00:17:04,420 --> 00:17:06,420 and ending point are the same. 286 00:17:06,420 --> 00:17:07,580 Hopefully found that useful. 287 00:17:07,579 --> 00:17:08,679 And we'll some examples with that. 288 00:17:08,680 --> 00:17:10,990 And actually in the next video I'll prove another interesting 289 00:17:10,990 --> 00:17:13,480 outcome based on this one.