1 00:00:00,000 --> 00:00:00,650 2 00:00:00,650 --> 00:00:03,529 Let's say I have a position vector function that looks like 3 00:00:03,529 --> 00:00:12,201 this. r of t is equal to x of t times the unit vector i plus y 4 00:00:12,201 --> 00:00:15,074 of t times the unit vector j. 5 00:00:15,074 --> 00:00:18,030 And let me actually graph this. 6 00:00:18,030 --> 00:00:20,580 So let's say, r of t, I want to draw it a little bit 7 00:00:20,579 --> 00:00:21,875 straighter than that. 8 00:00:21,875 --> 00:00:28,629 So that's my y-axis, that is my x-axis, and let's say r of t, 9 00:00:28,629 --> 00:00:30,899 and this is for t is less than, let me write this. 10 00:00:30,899 --> 00:00:35,810 So this is for t is between a and b. 11 00:00:35,810 --> 00:00:39,219 So when t is equal to a, we're at this vector right here. 12 00:00:39,219 --> 00:00:43,170 So if you actually substitute t is equal to a here, you'd get 13 00:00:43,170 --> 00:00:46,070 a position vector that would point to that point over there. 14 00:00:46,070 --> 00:00:49,670 And then, as t increases, it traces out a curve, or the 15 00:00:49,670 --> 00:00:52,210 endpoints of our position vectors trace a curve that 16 00:00:52,210 --> 00:00:53,730 looks something like that. 17 00:00:53,729 --> 00:00:58,189 So when t is equal to b, we get a position vector that points 18 00:00:58,189 --> 00:00:59,509 to that point right there. 19 00:00:59,509 --> 00:01:01,799 So this defines a path. 20 00:01:01,799 --> 00:01:05,569 And the path is going in this upward direction, 21 00:01:05,569 --> 00:01:06,949 just like that. 22 00:01:06,950 --> 00:01:10,769 Now let's say that we have any another position 23 00:01:10,769 --> 00:01:13,439 vector function. 24 00:01:13,439 --> 00:01:27,629 Let me call it r of t. 25 00:01:27,629 --> 00:01:28,219 It's a different one. 26 00:01:28,219 --> 00:01:29,969 It's the green r. 27 00:01:29,969 --> 00:01:31,349 r of t. 28 00:01:31,349 --> 00:01:35,949 Instead of being x of t times i, it's going to be x of a plus 29 00:01:35,950 --> 00:01:41,890 b minus t times i, and instead of y of t, it's going to be y 30 00:01:41,890 --> 00:01:46,469 of a plus b minus t times i. 31 00:01:46,469 --> 00:01:48,640 And we've seen this in the last two videos. 32 00:01:48,640 --> 00:01:54,310 This, the path defined by this position vector function is 33 00:01:54,310 --> 00:01:56,019 going look more like this. 34 00:01:56,019 --> 00:01:58,640 Let me draw my axes. 35 00:01:58,640 --> 00:02:01,820 This is my y-axis, that is my x-axis. 36 00:02:01,819 --> 00:02:06,429 Maybe I should label them, y and x. 37 00:02:06,430 --> 00:02:08,710 This path is going to look just like this. 38 00:02:08,710 --> 00:02:12,390 But instead of starting here and going there, when t is 39 00:02:12,389 --> 00:02:15,779 equal to 1, let me make it clear, this is also true for 40 00:02:15,780 --> 00:02:18,840 a is less than or equal to t, which is less than b. 41 00:02:18,840 --> 00:02:20,750 So t is going to go from a to b. 42 00:02:20,750 --> 00:02:22,900 But here, when t is equal to a, you substitute it 43 00:02:22,900 --> 00:02:25,659 over here, you're going to get this vector. 44 00:02:25,659 --> 00:02:30,520 You're going to start over there, and as you increment t, 45 00:02:30,520 --> 00:02:33,600 as you make it larger and larger and larger, you're going 46 00:02:33,599 --> 00:02:37,009 to trace out that same path, but in the opposite direction. 47 00:02:37,009 --> 00:02:39,909 48 00:02:39,909 --> 00:02:42,750 And so when t is equal to b, you put that in here, you're 49 00:02:42,750 --> 00:02:45,550 actually going to get x of a and y of a there, right, the 50 00:02:45,550 --> 00:02:48,300 b's cancel out, and so you're going to point right like that. 51 00:02:48,300 --> 00:02:50,480 So these are the same, you could imagine the shape of 52 00:02:50,479 --> 00:02:54,310 these paths are the same, but we're going in the exact 53 00:02:54,310 --> 00:02:55,750 opposite direction. 54 00:02:55,750 --> 00:02:58,180 So what we're going to do in this video is to see what 55 00:02:58,180 --> 00:03:02,390 happens, how, I guess you could say, if I have some vector 56 00:03:02,389 --> 00:03:17,009 field f of xy equals p of xy i plus q of xy j. 57 00:03:17,009 --> 00:03:17,219 Right? 58 00:03:17,219 --> 00:03:22,250 This is just a vector field over the x-y plane. 59 00:03:22,250 --> 00:03:26,800 How the line integral of this vector field, of this vector 60 00:03:26,800 --> 00:03:34,640 field over this path, compares to the line integral the same 61 00:03:34,639 --> 00:03:36,500 vector field over that path. 62 00:03:36,500 --> 00:03:40,280 How that compares to this. 63 00:03:40,280 --> 00:03:41,770 We'll call this the minus curve. 64 00:03:41,770 --> 00:03:46,020 So this is the positive curve, we're going to call 65 00:03:46,020 --> 00:03:47,909 this the minus curve. 66 00:03:47,909 --> 00:03:50,400 So how does it, going over the positive curve, compare 67 00:03:50,400 --> 00:03:55,330 to going over the minus curve? f of f dot dr. 68 00:03:55,330 --> 00:03:57,640 So before I break into the math, let's just think 69 00:03:57,639 --> 00:03:58,309 about a little bit. 70 00:03:58,310 --> 00:04:00,539 Let me draw this vector field f. 71 00:04:00,539 --> 00:04:03,719 So maybe it looks, I'm just going to draw random stuff. 72 00:04:03,719 --> 00:04:07,539 So you know, on every point in the x-y plane, it has a vector, 73 00:04:07,539 --> 00:04:11,560 it defines or maps a vector, to every point on the x-y plane. 74 00:04:11,560 --> 00:04:13,800 But we really care about the points that are on the curve. 75 00:04:13,800 --> 00:04:17,600 So maybe on the curve, you know, this is the vector field 76 00:04:17,600 --> 00:04:19,685 at the points on the curve. 77 00:04:19,685 --> 00:04:21,149 And let me draw it over here, too. 78 00:04:21,149 --> 00:04:23,519 So all the points on the curve where we care about, this is 79 00:04:23,519 --> 00:04:29,120 our vector field, that is our vector field. 80 00:04:29,120 --> 00:04:32,030 And let's just get an intuition of what's going to be going. 81 00:04:32,029 --> 00:04:35,989 We're summing over the dot, we're taking each point along 82 00:04:35,990 --> 00:04:41,439 the line, and we're summing, let me start over here. 83 00:04:41,439 --> 00:04:43,153 We're taking each point along the line, let me do it 84 00:04:43,153 --> 00:04:45,450 in a different color. 85 00:04:45,449 --> 00:04:49,034 And we're summing the dot product of the value of the 86 00:04:49,035 --> 00:04:53,910 vector field at that point, the dot product of that, with dr, 87 00:04:53,910 --> 00:04:57,430 or the differential of our position vector function. 88 00:04:57,430 --> 00:05:01,769 And dr, you can kind of imagine, as an infinitesimally 89 00:05:01,769 --> 00:05:04,620 small vector going in the direction of our movement. 90 00:05:04,620 --> 00:05:08,079 And we take this dot product here, it's essentially, it's 91 00:05:08,079 --> 00:05:10,339 going to be a scalar value, but the dot product, if you 92 00:05:10,339 --> 00:05:15,529 remember, it's the magnitude of f in the direction of dr, 93 00:05:15,529 --> 00:05:18,349 times the magnitude of dr. 94 00:05:18,350 --> 00:05:21,750 So it's this, you can imagine it's the shadow of f onto 95 00:05:21,750 --> 00:05:24,040 dr. Let me zoom into that, because I think it's useful. 96 00:05:24,040 --> 00:05:26,879 So this little thing that I'm drawing right here, let's 97 00:05:26,879 --> 00:05:29,069 say that this is my path. 98 00:05:29,069 --> 00:05:31,469 This is f at that point. 99 00:05:31,470 --> 00:05:34,260 f at that point looks something like that. 100 00:05:34,259 --> 00:05:37,500 And then dr at this point looks something like that. 101 00:05:37,500 --> 00:05:40,839 Let me do it in different color. dr looks 102 00:05:40,839 --> 00:05:42,229 something like that. 103 00:05:42,230 --> 00:05:43,930 So that is f. 104 00:05:43,930 --> 00:05:47,300 And so the dot product of these two says, ok, how much of f is 105 00:05:47,300 --> 00:05:48,670 going in the same direction as dr? 106 00:05:48,670 --> 00:05:50,939 And you can kind of imagine, there's a shadow. 107 00:05:50,939 --> 00:05:54,800 If you take the f that's going in the same direction as dr, 108 00:05:54,800 --> 00:05:58,410 the magnitude of that times the magnitude of dr, that 109 00:05:58,410 --> 00:05:59,400 is the dot product. 110 00:05:59,399 --> 00:06:01,169 In this case, we're going to get a positive number. 111 00:06:01,170 --> 00:06:03,360 Because this length is positive, this length is 112 00:06:03,360 --> 00:06:06,030 positive, that's going to be a positive number. 113 00:06:06,029 --> 00:06:08,179 Now what if our dr was going in the opposite direction, 114 00:06:08,180 --> 00:06:09,600 as it is in this case? 115 00:06:09,600 --> 00:06:12,445 So let me draw maybe that same part of the curve. 116 00:06:12,444 --> 00:06:15,360 117 00:06:15,360 --> 00:06:21,500 We have our f, our f will look something like that. 118 00:06:21,500 --> 00:06:23,519 I'm drawing this exact same part of the curve. 119 00:06:23,519 --> 00:06:27,469 But now our dr isn't going in that direction. 120 00:06:27,470 --> 00:06:31,470 Our dr that at this point is going to be going in 121 00:06:31,470 --> 00:06:32,730 the other direction. 122 00:06:32,730 --> 00:06:34,800 We're tracing the curve in the opposite direction. 123 00:06:34,800 --> 00:06:37,540 Our dr is now going to be going in that direction. 124 00:06:37,540 --> 00:06:41,110 125 00:06:41,110 --> 00:06:44,430 So if you do f dot dr, you're taking the shadow, or how much 126 00:06:44,430 --> 00:06:46,550 of f is going in the direction of dr, you take the shadow 127 00:06:46,550 --> 00:06:50,550 down here, it's going in the opposite direction of dr. So 128 00:06:50,550 --> 00:06:52,840 you can imagine that when you multiply the magnitudes, you 129 00:06:52,839 --> 00:06:53,929 should get a negative number. 130 00:06:53,930 --> 00:06:56,870 Our direction is now opposite, they're not going in the-- the 131 00:06:56,870 --> 00:07:01,319 shadow of f onto the same direction is dr is going in the 132 00:07:01,319 --> 00:07:03,540 opposite direction as dr. In this case, it's going in 133 00:07:03,540 --> 00:07:05,090 the same direction as dr. 134 00:07:05,089 --> 00:07:08,009 So the intuition is that maybe these two things are the 135 00:07:08,009 --> 00:07:09,480 negative of each other. 136 00:07:09,480 --> 00:07:12,490 And now we can do some math and try to see if 137 00:07:12,490 --> 00:07:16,090 that is definitely, definitely the case. 138 00:07:16,089 --> 00:07:20,169 So let us first figure out, let's write an expression for 139 00:07:20,170 --> 00:07:29,830 the differential dr. So in this case, dr, dr dt is going to be 140 00:07:29,829 --> 00:07:40,990 equal to x prime of t times i plus y prime of t times j. 141 00:07:40,990 --> 00:07:48,129 In this other example, in the reverse case, our dr, dr 142 00:07:48,129 --> 00:07:51,350 dt, is going to be, what's it going to be equal to? 143 00:07:51,350 --> 00:07:54,540 It's the derivative of x with respect to t. 144 00:07:54,540 --> 00:07:57,530 The derivative of this term with respect to t, that's the 145 00:07:57,529 --> 00:08:01,259 derivative of the inside, which is minus 1, or minus, times 146 00:08:01,259 --> 00:08:03,839 the derivative of the outside with respect to the inside. 147 00:08:03,839 --> 00:08:07,079 So that's going to be, derivative of the inside is 148 00:08:07,079 --> 00:08:10,349 minus 1, times the derivative of the outside with respect to 149 00:08:10,350 --> 00:08:15,260 the inside. x prime of a plus b minus t times i. 150 00:08:15,259 --> 00:08:17,639 And then same thing for the second term. 151 00:08:17,639 --> 00:08:20,930 Derivative of y of this term with respect to the inside, 152 00:08:20,930 --> 00:08:23,959 which is minus 1, times the derivative of the outside with 153 00:08:23,959 --> 00:08:27,469 respect to the inside, which is y prime of a plus b minus t. 154 00:08:27,470 --> 00:08:30,240 So this is going to be the derivative of the inside, 155 00:08:30,240 --> 00:08:36,649 times y prime of a plus b minus is t j. 156 00:08:36,649 --> 00:08:40,659 So this is dr dt in this case, this is dr dt in that case. 157 00:08:40,659 --> 00:08:45,620 And if we wanted to write the differential dr in the forward 158 00:08:45,620 --> 00:08:51,509 curve example, it's going to be equal to x prime of t times i 159 00:08:51,509 --> 00:08:58,870 plus y prime of t times j times the scalar dt. 160 00:08:58,870 --> 00:09:01,220 I could multiply it down into each of these terms, but 161 00:09:01,220 --> 00:09:04,060 it keeps it simple, just leaving it on the outside. 162 00:09:04,059 --> 00:09:06,149 Same logic over here. 163 00:09:06,149 --> 00:09:10,509 dr is equal to minus x. 164 00:09:10,509 --> 00:09:12,850 I changed my shade of green, but at least it's still green. 165 00:09:12,850 --> 00:09:21,100 a plus b minus t i minus y prime a plus b minus t j, 166 00:09:21,100 --> 00:09:25,320 and I'm multiplying both sides by dt. 167 00:09:25,320 --> 00:09:27,940 Now we're ready to express this as a function of t. 168 00:09:27,940 --> 00:09:31,870 So this curve right here, I'll do it in pink, the pink one is 169 00:09:31,870 --> 00:09:36,340 going to be equal to the integral from t is equal to a 170 00:09:36,340 --> 00:09:53,629 to t is equal to b of f of f of x of f of x of t y of t dot 171 00:09:53,629 --> 00:09:56,820 this thing over here, which is, I'll just write out here, I 172 00:09:56,820 --> 00:09:59,002 could simplify it later. 173 00:09:59,001 --> 00:10:06,259 x prime of t i plus y prime of t j. 174 00:10:06,259 --> 00:10:08,929 And then all of that times the scalar dt. 175 00:10:08,929 --> 00:10:11,229 This'll be a scalar value, and then we'll have another scalar 176 00:10:11,230 --> 00:10:13,190 value of dt over there. 177 00:10:13,190 --> 00:10:16,490 Now, what is this going to be equal to if I take 178 00:10:16,490 --> 00:10:18,629 this reverse integral? 179 00:10:18,629 --> 00:10:21,274 The reverse integral is going to be the integral from, I'm 180 00:10:21,274 --> 00:10:26,879 going to need a little more space, from a to b, of f of not 181 00:10:26,879 --> 00:10:34,220 x of t, but x of a plus b minus t y of a plus b minus t. 182 00:10:34,220 --> 00:10:36,550 I'm writing it small so I have some space. 183 00:10:36,549 --> 00:10:41,159 Dot, this is a vector, so dot this guy right here, dot dr. 184 00:10:41,159 --> 00:10:53,799 Dot minus x prime of a plus b minus t i, minus y prime of, 185 00:10:53,799 --> 00:10:55,109 I'm using up too much space. 186 00:10:55,110 --> 00:10:57,769 Let me scroll, go back a little bit. 187 00:10:57,769 --> 00:10:59,679 Actually, let me take it make it even simpler. 188 00:10:59,679 --> 00:11:01,689 Let me take this minus sign out of it. 189 00:11:01,690 --> 00:11:04,400 Let me put a plus, and then I'll put the 190 00:11:04,399 --> 00:11:06,789 minus sign out front. 191 00:11:06,789 --> 00:11:09,259 So the minus sign is just a scalar value, so we could put 192 00:11:09,259 --> 00:11:11,710 that minus sign out, you know, when you take a dot product, 193 00:11:11,710 --> 00:11:15,180 and if you multiply a scalar times a dot product, you could 194 00:11:15,179 --> 00:11:17,679 just take the scalar out, that's all I'm saying. 195 00:11:17,679 --> 00:11:20,879 So we take that minus sign out to this part right here. 196 00:11:20,879 --> 00:11:30,029 And then you have x prime of a plus b minus t i plus y prime 197 00:11:30,029 --> 00:11:39,970 of a plus b minus t, scroll over a little bit, t j dt. 198 00:11:39,970 --> 00:11:43,129 So the this is the forward, this is when we're following it 199 00:11:43,129 --> 00:11:45,929 along the forward curve, this is when we're following it 200 00:11:45,929 --> 00:11:48,489 along the reverse curve. 201 00:11:48,490 --> 00:11:51,230 Now like we did with the scalar example, let's 202 00:11:51,230 --> 00:11:51,990 make a substitution. 203 00:11:51,990 --> 00:11:53,100 I want to make it very clear what I did. 204 00:11:53,100 --> 00:11:55,330 All I did here, is I just took the dot product, 205 00:11:55,330 --> 00:11:57,600 but this negative sign, I just took it out. 206 00:11:57,600 --> 00:12:00,290 I just said, this is the same thing as negative 1 times this 207 00:12:00,289 --> 00:12:02,199 thing, or negative 1 times this thing is the same 208 00:12:02,200 --> 00:12:03,230 thing as that. 209 00:12:03,230 --> 00:12:05,610 So let's make a substitution on this side, because I really 210 00:12:05,610 --> 00:12:08,899 just want to show you that this is the negative of that, right 211 00:12:08,899 --> 00:12:10,879 there, because that's what our intuition was going for. 212 00:12:10,879 --> 00:12:13,220 So let me just focus on that side. 213 00:12:13,220 --> 00:12:15,980 So let me make a substitution. 214 00:12:15,980 --> 00:12:21,529 u is equal to a plus b minus t. 215 00:12:21,529 --> 00:12:24,720 Then we get du is equal to minus dt, right? 216 00:12:24,720 --> 00:12:26,769 Just take the derivative of both sides. 217 00:12:26,769 --> 00:12:31,230 Or you get dt is equal to minus du. 218 00:12:31,230 --> 00:12:37,889 And then you get, when t is equal to a, u is equal 219 00:12:37,889 --> 00:12:40,049 to a plus b minus a. 220 00:12:40,049 --> 00:12:43,069 So then, u is equal to b. 221 00:12:43,070 --> 00:12:49,140 And then when t is equal to b, u is equal to a, right? 222 00:12:49,139 --> 00:12:52,529 Which is equal to b, a plus b minus b is a. 223 00:12:52,529 --> 00:12:53,230 u is equal to a. 224 00:12:53,230 --> 00:12:57,812 So this thing, using that substitution simplifies to, and 225 00:12:57,812 --> 00:13:03,379 this is the whole point, that simplifies to minus integral 226 00:13:03,379 --> 00:13:07,009 from u is, when t is a, u is b. 227 00:13:07,009 --> 00:13:09,569 From b, when t is b, u is a. 228 00:13:09,570 --> 00:13:13,930 The integral from u is equal to b to u is equal to a of 229 00:13:13,929 --> 00:13:21,139 f of x of u y of u, right? 230 00:13:21,139 --> 00:13:24,000 That is u, that is u. 231 00:13:24,000 --> 00:13:39,330 Dot x prime of u times i, that's u right there, plus 232 00:13:39,330 --> 00:13:44,240 y prime of u times j. 233 00:13:44,240 --> 00:13:47,360 And then, instead of a dt, I need to put a du. 234 00:13:47,360 --> 00:13:49,970 dt is equal to minus du. 235 00:13:49,970 --> 00:13:53,190 So I could write minus du here, or just to not make it 236 00:13:53,190 --> 00:13:55,780 confusing, I'll put the du here, and take the 237 00:13:55,779 --> 00:13:56,899 minus out front. 238 00:13:56,899 --> 00:14:00,230 I already have a minus out there, so they cancel out. 239 00:14:00,230 --> 00:14:03,730 They will cancel out, just like that. 240 00:14:03,730 --> 00:14:06,480 And so you might say, hey Sal, these two things look pretty 241 00:14:06,480 --> 00:14:07,320 similar to each other. 242 00:14:07,320 --> 00:14:08,770 They don't like they're negative of each other. 243 00:14:08,769 --> 00:14:12,079 And I'd say, well, you're almost right, except this guy's 244 00:14:12,080 --> 00:14:15,240 limits of integration are reversed from this guy. 245 00:14:15,240 --> 00:14:17,810 So this thing right here, if we reverse the limits of 246 00:14:17,809 --> 00:14:20,729 integration, we have to then make it negative. 247 00:14:20,730 --> 00:14:29,389 So this is equal to minus the integral from a to b of the 248 00:14:29,389 --> 00:14:40,919 vector f of x of u y of u dot x prime of u i plus 249 00:14:40,919 --> 00:14:45,539 y prime of u j du. 250 00:14:45,539 --> 00:14:47,899 And now this is identical. 251 00:14:47,899 --> 00:14:50,509 This integral, this definite integral, is identical to 252 00:14:50,509 --> 00:14:51,620 that definite integral. 253 00:14:51,620 --> 00:14:52,870 We just have a different variable. 254 00:14:52,870 --> 00:14:55,620 We're doing dt here, we have du here, but we're going to get 255 00:14:55,620 --> 00:14:59,220 the same exact number for any a or b, and given this 256 00:14:59,220 --> 00:15:06,050 vector f and the position vector path r of t. 257 00:15:06,049 --> 00:15:11,429 So just to summarize everything up, when you're dealing with 258 00:15:11,429 --> 00:15:16,500 line integrals over vector fields, the direction matters. 259 00:15:16,500 --> 00:15:18,529 If you go in the reverse direction, you're going to get 260 00:15:18,529 --> 00:15:20,519 the negative version of that. 261 00:15:20,519 --> 00:15:23,610 And that's because at any point we take the dot product, you're 262 00:15:23,610 --> 00:15:27,060 not going in the, necessarily, you're going in the opposite 263 00:15:27,059 --> 00:15:29,329 direction, so it'll be the negative of each other. 264 00:15:29,330 --> 00:15:31,730 But when you're dealing with the scalar field, we saw on the 265 00:15:31,730 --> 00:15:37,430 last video, we saw that it doesn't matter which direction 266 00:15:37,429 --> 00:15:39,099 that you traverse the path in. 267 00:15:39,100 --> 00:15:41,490 That the positive path has the same value as 268 00:15:41,490 --> 00:15:42,210 the negative path. 269 00:15:42,210 --> 00:15:44,600 And that's just because we're just trying to find the 270 00:15:44,600 --> 00:15:46,590 area of that curtain. 271 00:15:46,590 --> 00:15:49,670 Hopefully you found that mildly amusing. 272 00:15:49,669 --> 00:15:50,265