1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:04,140 In the last video, we finished off with these two results. 3 00:00:04,139 --> 00:00:06,849 We started off just thinking about what it means to take 4 00:00:06,849 --> 00:00:10,000 the partial derivative of vector-valued function, and I 5 00:00:10,000 --> 00:00:13,000 got to these kind of, you might call them, bizarre results. 6 00:00:13,000 --> 00:00:15,230 You know, what was the whole point in getting here, Sal? 7 00:00:15,230 --> 00:00:18,820 And the whole point is so that I can give you the tools you 8 00:00:18,820 --> 00:00:21,839 need to understand what a surface integral is. 9 00:00:21,839 --> 00:00:27,320 So let's just think about, let's draw the st plane, 10 00:00:27,320 --> 00:00:29,640 and then see how it gets transformed into 11 00:00:29,640 --> 00:00:31,719 this surface r. 12 00:00:31,719 --> 00:00:33,619 So let's do that. 13 00:00:33,619 --> 00:00:36,489 14 00:00:36,490 --> 00:00:43,760 Let's say that is the t-axis, and let's say that that is the 15 00:00:43,759 --> 00:00:49,239 s-axis, and let's say that our vector-valued function, our 16 00:00:49,240 --> 00:00:52,550 positioned vector-valued function, is defined from s's 17 00:00:52,549 --> 00:00:57,589 between a and b, I'm just picking arbitrary boundaries, 18 00:00:57,590 --> 00:01:01,550 and between t being equal to c and d. 19 00:01:01,549 --> 00:01:04,250 20 00:01:04,250 --> 00:01:09,319 So the area under question, if you take any t and any s in 21 00:01:09,319 --> 00:01:15,709 this rectangle right here, it will be mapped to part 22 00:01:15,709 --> 00:01:16,890 of that surface. 23 00:01:16,890 --> 00:01:19,099 And if you map each of these points, you will eventually 24 00:01:19,099 --> 00:01:21,129 get the surface r. 25 00:01:21,129 --> 00:01:25,250 So let me draw r in 3 dimensions. 26 00:01:25,250 --> 00:01:26,769 A surface in 3D. 27 00:01:26,769 --> 00:01:33,090 So that is our x-axis, that is our y-axis, and then 28 00:01:33,090 --> 00:01:36,930 that is the z-axis. 29 00:01:36,930 --> 00:01:40,780 And just as a bit of a reminder, it might look 30 00:01:40,780 --> 00:01:42,370 something like this. 31 00:01:42,370 --> 00:01:46,570 If we were to, this point right here, where s is equal to a, 32 00:01:46,569 --> 00:01:51,559 and t is equal to c, remember, we're going to draw the surface 33 00:01:51,560 --> 00:01:54,540 indicated by the position vector function 34 00:01:54,540 --> 00:01:57,380 s, r of s and t. 35 00:01:57,379 --> 00:02:01,229 So this point right here, when s is a and t is c, maybe it 36 00:02:01,230 --> 00:02:04,120 maps to, I'm just, you know, that point! 37 00:02:04,120 --> 00:02:05,070 Right there. 38 00:02:05,069 --> 00:02:09,539 When you take a and c, and you put it into this thing over 39 00:02:09,539 --> 00:02:12,870 here, you're just going to get the vector that points at that. 40 00:02:12,870 --> 00:02:15,840 So you could say, it'll give you a position vector 41 00:02:15,840 --> 00:02:19,170 that'll point right at that position, right there. 42 00:02:19,169 --> 00:02:23,406 And then, let's say that this line, right here, if we were to 43 00:02:23,407 --> 00:02:31,570 hold s constant at a, and if we were to just vary t from c 44 00:02:31,569 --> 00:02:36,810 to d, maybe that looks something like this. 45 00:02:36,810 --> 00:02:40,080 I'm just drawing some arbitrary contour there. 46 00:02:40,080 --> 00:02:45,980 Maybe if we hold t constant at c, and vary s from a to b, 47 00:02:45,979 --> 00:02:50,209 maybe that'll look at something like that. 48 00:02:50,210 --> 00:02:50,980 I don't know. 49 00:02:50,979 --> 00:02:53,979 I'm just trying to show you an example. 50 00:02:53,979 --> 00:02:58,149 So this point right here would correspond to that point right 51 00:02:58,150 --> 00:03:03,379 there, when you put it into the vector-valued function r, you 52 00:03:03,379 --> 00:03:07,419 would get a vector that points to that point, just like that. 53 00:03:07,419 --> 00:03:11,709 And this point right here in purple, when you evaluate r of 54 00:03:11,710 --> 00:03:16,659 s and t, it'll give you a vector that points right there, 55 00:03:16,659 --> 00:03:18,849 to that point over there, and we could do a couple of other 56 00:03:18,849 --> 00:03:21,430 points, just to get an idea of what the surface looks like, 57 00:03:21,430 --> 00:03:24,390 although I'm trying to keep things as general as possible. 58 00:03:24,389 --> 00:03:30,729 So maybe, let me do it in this bluish color. 59 00:03:30,729 --> 00:03:37,349 This, if we hold p at d and vary s from a to b, we're 60 00:03:37,349 --> 00:03:38,229 going to start here. 61 00:03:38,229 --> 00:03:41,539 This is when t is d and s is a. 62 00:03:41,539 --> 00:03:44,879 And when you vary it, maybe you get something like that. 63 00:03:44,879 --> 00:03:45,319 I don't know. 64 00:03:45,319 --> 00:03:50,900 So this point right here would correspond to a vector that 65 00:03:50,900 --> 00:03:56,430 points to that point, right there, and then finally this 66 00:03:56,430 --> 00:04:02,599 line, or this, if we hold s at b and vary t between c and d, 67 00:04:02,599 --> 00:04:08,479 we're going to go from that point, to that point. 68 00:04:08,479 --> 00:04:12,139 So it's going to look something like this-- oh, sorry, we're 69 00:04:12,139 --> 00:04:14,629 going to go from this point to that point. 70 00:04:14,629 --> 00:04:18,579 We're holding s at b, varying t from c to d, maybe it 71 00:04:18,579 --> 00:04:21,289 looks something like that. 72 00:04:21,290 --> 00:04:26,200 So our surface, we went from this nice rectangular area in 73 00:04:26,199 --> 00:04:30,250 the ts plane, and it gets transformed into this 74 00:04:30,250 --> 00:04:31,879 wacky-looking surface. 75 00:04:31,879 --> 00:04:34,920 We could even draw some other things right here. 76 00:04:34,920 --> 00:04:40,150 Let's say we get some arbitrary value. 77 00:04:40,149 --> 00:04:42,719 Let me pick a new color, I'll do it in white. 78 00:04:42,720 --> 00:04:44,430 Or a new noncolor. 79 00:04:44,430 --> 00:04:46,860 And let's say if we hold s at that constant value and we 80 00:04:46,860 --> 00:04:51,220 vary t, maybe that will look for something like this. 81 00:04:51,220 --> 00:04:55,690 Maybe that we'll look something like, well, I don't know. 82 00:04:55,689 --> 00:04:58,180 Maybe it'll look something like that. 83 00:04:58,180 --> 00:05:01,519 So you get an idea of what the surface might look like. 84 00:05:01,519 --> 00:05:04,139 Now, given this, I want to think about what 85 00:05:04,139 --> 00:05:06,349 these quantities are. 86 00:05:06,350 --> 00:05:08,830 And then when we visualize what these quantities are, we'll be 87 00:05:08,829 --> 00:05:11,629 able to kind of use these results of the last videos 88 00:05:11,629 --> 00:05:15,159 to do something that I think will be useful. 89 00:05:15,160 --> 00:05:19,330 So let's say that we pick arbitrary s and t. 90 00:05:19,329 --> 00:05:23,859 So this is the point, let me just we pick it right here. 91 00:05:23,860 --> 00:05:27,139 92 00:05:27,139 --> 00:05:30,019 That is the point s, t. 93 00:05:30,019 --> 00:05:31,089 s comma t. 94 00:05:31,089 --> 00:05:36,389 If you were to put those values in here maybe it maps to, and I 95 00:05:36,389 --> 00:05:38,849 want to make sure I'm consistent with everything I've 96 00:05:38,850 --> 00:05:43,720 drawn, maybe it maps to this point right here. 97 00:05:43,720 --> 00:05:46,430 Maybe it maps to that point right there. 98 00:05:46,430 --> 00:05:52,620 So this point right here, that is r of s and t. 99 00:05:52,620 --> 00:05:53,939 For a particular s and t. 100 00:05:53,939 --> 00:05:55,649 I mean, I could put little subscripts here, but 101 00:05:55,649 --> 00:05:56,789 I want to be general. 102 00:05:56,790 --> 00:05:59,069 I could call this a, well, I already used a and b. 103 00:05:59,069 --> 00:06:02,009 I could call this x and y, this would be r of x and y. 104 00:06:02,009 --> 00:06:05,819 It would map to that point, right there. 105 00:06:05,819 --> 00:06:10,089 So that right there, or that right there. 106 00:06:10,089 --> 00:06:13,500 Now let's see what happens if we take, if we move 107 00:06:13,500 --> 00:06:15,790 just in the s direction. 108 00:06:15,790 --> 00:06:20,270 So this is we could do that as s. 109 00:06:20,269 --> 00:06:24,310 Now let us move forward by a differential, by a 110 00:06:24,310 --> 00:06:26,310 super small amount of s. 111 00:06:26,310 --> 00:06:30,829 So this right here, let's call that a s plus a super 112 00:06:30,829 --> 00:06:33,149 small differential in s. 113 00:06:33,149 --> 00:06:33,909 That's right there. 114 00:06:33,910 --> 00:06:35,850 So that point. 115 00:06:35,850 --> 00:06:39,430 Let me do that in a better color, in this yellow. 116 00:06:39,430 --> 00:06:42,319 So that point right there is the point s plus 117 00:06:42,319 --> 00:06:44,079 my differential of s. 118 00:06:44,079 --> 00:06:46,550 I could write delta s, but I wanted a super small 119 00:06:46,550 --> 00:06:49,560 change in s, comma t. 120 00:06:49,560 --> 00:06:52,490 And what is that going to get mapped to? 121 00:06:52,490 --> 00:06:55,379 Well, if we apply these two point in r, we're going to 122 00:06:55,379 --> 00:07:01,659 get something that maybe is right over there. 123 00:07:01,660 --> 00:07:02,570 And I want to be very clear. 124 00:07:02,569 --> 00:07:11,189 This right here, that is r of s plus ds comma t. 125 00:07:11,189 --> 00:07:11,850 That's what that is. 126 00:07:11,850 --> 00:07:15,520 That's the point when we just shift s by a super small 127 00:07:15,519 --> 00:07:19,000 differential, this distance here, you can view as ds. 128 00:07:19,000 --> 00:07:21,399 It's a super small change in s. 129 00:07:21,399 --> 00:07:24,929 And then when we map it or transform it, or put that 130 00:07:24,930 --> 00:07:29,180 point into our vector-valued function, let me copy and paste 131 00:07:29,180 --> 00:07:32,600 the original vector-valued function, just so we have a 132 00:07:32,600 --> 00:07:34,900 good image of what we're talking about this whole time. 133 00:07:34,899 --> 00:07:41,079 134 00:07:41,079 --> 00:07:44,620 Let me put it right down there. 135 00:07:44,620 --> 00:07:51,000 So just to be clear what's going on, when we took this 136 00:07:51,000 --> 00:07:56,160 little blue point right here, this s and t, and we put the s 137 00:07:56,160 --> 00:08:00,480 and t values here, we get a vector that points to that 138 00:08:00,480 --> 00:08:02,560 point on the surface, right there. 139 00:08:02,560 --> 00:08:06,769 When you add a ds to your s-values, you get a vector 140 00:08:06,769 --> 00:08:09,659 that points at that yellow point right there. 141 00:08:09,660 --> 00:08:12,570 142 00:08:12,569 --> 00:08:16,694 So going back to the results we got in the last presentation, 143 00:08:16,694 --> 00:08:18,639 or the last video, what is this? 144 00:08:18,639 --> 00:08:22,966 r of s plus delta s, or r of s plus ds, the differential of 145 00:08:22,966 --> 00:08:27,250 s, t, well, that is that, right there. 146 00:08:27,250 --> 00:08:31,019 That is the vector that points to that position. 147 00:08:31,019 --> 00:08:33,189 This right here is a vector that points to this 148 00:08:33,190 --> 00:08:34,480 blue position. 149 00:08:34,480 --> 00:08:36,759 So what is the difference of those two vectors? 150 00:08:36,759 --> 00:08:40,429 And this is a bit of basic vector math, 151 00:08:40,429 --> 00:08:41,059 you might remember. 152 00:08:41,059 --> 00:08:43,909 The difference of these two vectors, head to tails. 153 00:08:43,909 --> 00:08:45,829 The difference of these two vectors is going 154 00:08:45,830 --> 00:08:47,020 to be this vector. 155 00:08:47,019 --> 00:08:49,449 If you subtract this vector that vector, you're going to 156 00:08:49,450 --> 00:08:51,190 get that vector, right there. 157 00:08:51,190 --> 00:08:54,695 158 00:08:54,695 --> 00:08:57,650 A vector that looks just like that. 159 00:08:57,649 --> 00:09:01,379 So that's what this is equal to, that vector. 160 00:09:01,379 --> 00:09:02,340 And it makes sense. 161 00:09:02,340 --> 00:09:06,450 This blue vector plus the orange vector, this vector, 162 00:09:06,450 --> 00:09:08,120 right here, plus the orange vector, is equal 163 00:09:08,120 --> 00:09:08,754 to this vector. 164 00:09:08,754 --> 00:09:09,860 It makes complete sense. 165 00:09:09,860 --> 00:09:11,080 Heads to tails. 166 00:09:11,080 --> 00:09:12,650 So that's what that represents. 167 00:09:12,649 --> 00:09:14,625 Now let's do the same thing in the t direction. 168 00:09:14,625 --> 00:09:17,879 169 00:09:17,879 --> 00:09:19,019 I'm running out of color. 170 00:09:19,019 --> 00:09:21,559 171 00:09:21,559 --> 00:09:24,799 I'll go back to the pink, or maybe the magenta. 172 00:09:24,799 --> 00:09:26,609 So we had that s and t. 173 00:09:26,610 --> 00:09:32,960 Now if we go up a little bit, in that direction, let's say 174 00:09:32,960 --> 00:09:38,360 that that is t, so this is the point s, t plus a super small 175 00:09:38,360 --> 00:09:40,909 change in t, that's that point right there. 176 00:09:40,909 --> 00:09:46,179 This distance right there is dt, you can view it that way. 177 00:09:46,179 --> 00:09:51,279 If you put s and t plus dt into our vector value function, 178 00:09:51,279 --> 00:09:52,179 what are you to get? 179 00:09:52,179 --> 00:09:54,699 You're going to get a vector that maybe points to 180 00:09:54,700 --> 00:09:57,390 this point, right here. 181 00:09:57,389 --> 00:09:58,299 Maybe I'll draw it right here. 182 00:09:58,299 --> 00:10:00,689 Maybe it points to this point, right here. 183 00:10:00,690 --> 00:10:02,660 A vector that points right there. 184 00:10:02,659 --> 00:10:05,909 So that will be mapped to a vector that points to that 185 00:10:05,909 --> 00:10:07,549 position, right over there. 186 00:10:07,549 --> 00:10:11,859 Now by the same argument that we did on the s-side, this 187 00:10:11,860 --> 00:10:14,909 point, or the vector that points to that, that 188 00:10:14,909 --> 00:10:19,429 is r of st plus dt. 189 00:10:19,429 --> 00:10:24,250 That is the exact same thing as that right there, and of 190 00:10:24,250 --> 00:10:26,940 course, this, we already saw. 191 00:10:26,940 --> 00:10:29,800 This is the same thing as that over there. 192 00:10:29,799 --> 00:10:33,370 So what is that vector minus this blue vector? 193 00:10:33,370 --> 00:10:36,830 The magenta vector minus the blue vector? 194 00:10:36,830 --> 00:10:40,100 Well, once again, this is hopefully a bit of a 195 00:10:40,100 --> 00:10:41,960 review of adding vectors. 196 00:10:41,960 --> 00:10:44,600 It's going to be a vector that looks like this. 197 00:10:44,600 --> 00:10:45,629 I'll do it in white. 198 00:10:45,629 --> 00:10:47,519 It's going to be a vector that looks like that. 199 00:10:47,519 --> 00:10:52,309 200 00:10:52,309 --> 00:10:54,739 And you can imagine, if you take the blue vector plus the 201 00:10:54,740 --> 00:10:57,210 white vector, the blue vector plus this white vector is going 202 00:10:57,210 --> 00:10:58,320 to equal this purple vector. 203 00:10:58,320 --> 00:11:00,910 So it makes sense, if the purple vector minus the blue 204 00:11:00,909 --> 00:11:04,189 vector is going to be equal to this white vector. 205 00:11:04,190 --> 00:11:06,330 So something interesting is going on here. 206 00:11:06,330 --> 00:11:10,020 I have these two, this is a vector that is kind of going 207 00:11:10,019 --> 00:11:15,759 along this parameterized surface, as we changed our 208 00:11:15,759 --> 00:11:17,590 s by a super small amount. 209 00:11:17,590 --> 00:11:20,700 And then this is a vector that is going along our surface if 210 00:11:20,700 --> 00:11:25,030 we change our t by a super small amount. 211 00:11:25,029 --> 00:11:28,189 Now, you may or may not remember this, and I've 212 00:11:28,190 --> 00:11:31,280 done several videos where I showed this to you. 213 00:11:31,279 --> 00:11:35,500 But the magnitude, if I take 2 vectors, and I take their cross 214 00:11:35,500 --> 00:11:40,370 product, so if I take the cross product of a and b, and I take 215 00:11:40,370 --> 00:11:42,870 the magnitude of the resulting vector-- remember, when you 216 00:11:42,870 --> 00:11:45,870 take the cross product, you get a third vector that is 217 00:11:45,870 --> 00:11:47,529 perpendicular to both of these. 218 00:11:47,529 --> 00:11:51,029 But if you were just to take the magnitude of that vector, 219 00:11:51,029 --> 00:12:04,279 that is equal to the area of a parallelogram, 220 00:12:04,279 --> 00:12:09,509 defined by a and b. 221 00:12:09,509 --> 00:12:11,289 What do I mean by that? 222 00:12:11,289 --> 00:12:20,939 Well, if that is vector a and that is vector b, that's a and 223 00:12:20,940 --> 00:12:24,400 that is b, if you were to just take the cross product of those 224 00:12:24,399 --> 00:12:26,289 two, you're going to get a third vector. 225 00:12:26,289 --> 00:12:27,879 You're going to get a third vector that's perpendicular 226 00:12:27,879 --> 00:12:30,659 to both of them, it kind of popped out of the page. 227 00:12:30,659 --> 00:12:33,559 That would be a cross b. 228 00:12:33,559 --> 00:12:37,049 But the magnitude of this, so if you just take a cross 229 00:12:37,049 --> 00:12:38,639 product, you're going to get a vector. 230 00:12:38,639 --> 00:12:41,210 But then if you take the magnitude of that vector, 231 00:12:41,210 --> 00:12:43,300 you're just saying, how big is that vector, how 232 00:12:43,299 --> 00:12:44,689 long is that vector? 233 00:12:44,690 --> 00:12:47,160 That's going to be the area of the parallelogram 234 00:12:47,159 --> 00:12:48,689 defined by a and b. 235 00:12:48,690 --> 00:12:50,650 And I've proved that in the linear algebra videos, maybe 236 00:12:50,649 --> 00:12:51,899 I'll prove it again in this. 237 00:12:51,899 --> 00:12:57,689 I mean it's because-- well, I won't go into that in detail. 238 00:12:57,690 --> 00:13:00,290 I've done it before, don't want to make this video too long. 239 00:13:00,289 --> 00:13:03,899 So the parallelogram defined by a and b, you just imagine a, 240 00:13:03,899 --> 00:13:07,399 and then take another kind of parallel version of a, is right 241 00:13:07,399 --> 00:13:10,230 over there, and another parallel version of b 242 00:13:10,230 --> 00:13:11,950 is right over there. 243 00:13:11,950 --> 00:13:16,980 So this is the parallelogram defined by a and b. 244 00:13:16,980 --> 00:13:20,909 So, going back to our surface example, if we were to take the 245 00:13:20,909 --> 00:13:25,069 cross product of this orange vector and this white vector, 246 00:13:25,070 --> 00:13:28,720 I'm going to get the surface area, I'm going to get the area 247 00:13:28,720 --> 00:13:32,580 of the parallelogram, defined by these two vectors. 248 00:13:32,580 --> 00:13:34,759 So if I take the parallel to that one, it will look 249 00:13:34,759 --> 00:13:38,799 something like this, and then a parallel to the orange one it 250 00:13:38,799 --> 00:13:41,569 will look something like that. 251 00:13:41,570 --> 00:13:45,360 So if I take the cross product of that and that, I'm going to 252 00:13:45,360 --> 00:13:49,879 get the area of that parallelogram. 253 00:13:49,879 --> 00:13:51,620 Now you might say, this is a surface, you're taking a 254 00:13:51,620 --> 00:13:53,250 straight-up parallelogram, but remember, these are 255 00:13:53,250 --> 00:13:55,169 super small changes. 256 00:13:55,169 --> 00:13:58,409 So you can imagine, a surface can be broken up into 257 00:13:58,409 --> 00:14:03,240 super small changes in parallelograms, or infinitely 258 00:14:03,240 --> 00:14:04,750 many parallelograms. 259 00:14:04,750 --> 00:14:08,129 And the more parallelograms you have, the better approximation 260 00:14:08,129 --> 00:14:10,159 of the surface you're going to have. 261 00:14:10,159 --> 00:14:13,559 And this is no different than when we first took integrals. 262 00:14:13,559 --> 00:14:15,689 We approximated the area under the curve with a 263 00:14:15,690 --> 00:14:17,600 bunch of rectangles. 264 00:14:17,600 --> 00:14:20,050 The more rectangles we had, the better. 265 00:14:20,049 --> 00:14:28,189 So let's call this little change in our surface d sigma, 266 00:14:28,190 --> 00:14:31,660 for a little change, for a little amount of our surface. 267 00:14:31,659 --> 00:14:33,819 And we could even say that, you know, the surface area of the 268 00:14:33,820 --> 00:14:36,580 surface will be the infinite sum of all of these 269 00:14:36,580 --> 00:14:39,070 infinitely small d sigmas. 270 00:14:39,070 --> 00:14:40,700 And there's actually a little notation for that. 271 00:14:40,700 --> 00:14:48,750 So surface area is equal to, we could integrate over the 272 00:14:48,750 --> 00:14:52,950 surface, and the notation usually is a capital sigma for 273 00:14:52,950 --> 00:14:56,080 a surface as opposed to a region or-- so you're 274 00:14:56,080 --> 00:14:57,970 integrating over the surface, and you do a double integral, 275 00:14:57,970 --> 00:14:59,490 because you're going in two directions, right? 276 00:14:59,490 --> 00:15:02,450 A surface is a kind of a folded, two-dimensional 277 00:15:02,450 --> 00:15:03,810 structure. 278 00:15:03,809 --> 00:15:05,309 And you're going to take the infinite sum of 279 00:15:05,309 --> 00:15:06,379 all of the d sigmas. 280 00:15:06,379 --> 00:15:09,689 281 00:15:09,690 --> 00:15:11,460 So this would be the surface area. 282 00:15:11,460 --> 00:15:13,110 So that's what a d sigma is. 283 00:15:13,110 --> 00:15:15,950 Now we just figured out, we just said, well, that d sigma 284 00:15:15,950 --> 00:15:20,050 can be represented, that value, that area, of that little part 285 00:15:20,049 --> 00:15:22,209 of the surface, of that parallelogram, can be 286 00:15:22,210 --> 00:15:25,800 represented as a cross product of those two vectors. 287 00:15:25,799 --> 00:15:26,549 So let me write here. 288 00:15:26,549 --> 00:15:28,839 And this is not rigorous mathematics. 289 00:15:28,840 --> 00:15:30,930 The whole point here is to give you the intuition of what a 290 00:15:30,929 --> 00:15:32,709 surface integral is all about. 291 00:15:32,710 --> 00:15:40,670 So we can write that d sigma is equal to the cross product of 292 00:15:40,669 --> 00:15:42,939 the orange vector and the white vector. 293 00:15:42,940 --> 00:15:45,270 The orange vector is this, but we could also 294 00:15:45,269 --> 00:15:45,860 write it like this. 295 00:15:45,860 --> 00:15:47,840 This was the result from the last video. 296 00:15:47,840 --> 00:15:49,460 I'll write it in orange. 297 00:15:49,460 --> 00:15:57,960 So the partial of r with respect to, I'm running out of 298 00:15:57,960 --> 00:16:04,400 space, with respect to s, ds, and it's going-- well, d sigma 299 00:16:04,399 --> 00:16:07,120 is going to be the magnitude of the cross product, not 300 00:16:07,120 --> 00:16:07,860 just the cross product. 301 00:16:07,860 --> 00:16:09,705 The cross product by itself will just give you a vector, 302 00:16:09,705 --> 00:16:11,930 and that's going to be useful when we start doing 303 00:16:11,929 --> 00:16:13,799 vector-valued surface integrals, but just think 304 00:16:13,799 --> 00:16:14,359 about it this way. 305 00:16:14,360 --> 00:16:17,399 So this orange vector is the same thing is that. 306 00:16:17,399 --> 00:16:19,679 And we're going to take the cross product of that 307 00:16:19,679 --> 00:16:21,279 with this white vector. 308 00:16:21,279 --> 00:16:24,829 This white vector is the same thing as that, which we saw, 309 00:16:24,830 --> 00:16:26,870 which was the same thing as this. 310 00:16:26,870 --> 00:16:28,899 The partial of r with respect to t, dt. 311 00:16:28,899 --> 00:16:35,980 312 00:16:35,980 --> 00:16:38,430 And we saw, if we take the magnitude of that, that's going 313 00:16:38,429 --> 00:16:43,629 to be equal to our little small change in area, or the 314 00:16:43,629 --> 00:16:47,500 area of this little parallelogram over here 315 00:16:47,500 --> 00:16:50,279 Now, you may or may not remember that if you take 316 00:16:50,279 --> 00:16:52,389 these, so let's just be clear. 317 00:16:52,389 --> 00:16:55,490 This and this, these are vectors, right? 318 00:16:55,490 --> 00:16:57,480 When you take the partial derivative of a vector-valued 319 00:16:57,480 --> 00:16:59,560 function, you're still getting a vector. 320 00:16:59,559 --> 00:17:01,399 This ds, this is a number. 321 00:17:01,399 --> 00:17:02,809 That's a number and that's a number. 322 00:17:02,809 --> 00:17:05,599 And you might remember, when we, in the linear algebra or 323 00:17:05,599 --> 00:17:08,959 whenever you first saw taking cross products, taking the 324 00:17:08,960 --> 00:17:10,850 cross product of some scalar multiple. 325 00:17:10,849 --> 00:17:12,669 You can take the scalars out. 326 00:17:12,670 --> 00:17:14,870 So if we take this number and that number, we essentially 327 00:17:14,869 --> 00:17:17,719 factor them out of the cross product. 328 00:17:17,720 --> 00:17:22,190 This is going to be equal to the magnitude of the cross 329 00:17:22,190 --> 00:17:30,279 product of the partial of r, with respect to s, crossed with 330 00:17:30,279 --> 00:17:35,750 the partial of r with respect to t, and then all of 331 00:17:35,750 --> 00:17:38,369 that times these two guys, over here. 332 00:17:38,369 --> 00:17:42,819 Times ds and dt. 333 00:17:42,819 --> 00:17:44,919 So I wrote this here, hey, maybe our surface area, if we 334 00:17:44,920 --> 00:17:47,830 were to take the sum of all of these little d sigmas, but 335 00:17:47,829 --> 00:17:49,879 there's no obvious way to evaluate that. 336 00:17:49,880 --> 00:17:53,980 But we know that all of the d sigmas, they're the same thing 337 00:17:53,980 --> 00:17:56,500 as if you take all of the ds's and all of the dt's. 338 00:17:56,500 --> 00:17:58,890 So you take all of the ds's, all of the dt's. 339 00:17:58,890 --> 00:18:01,240 So this is a ds times a dt, right? 340 00:18:01,240 --> 00:18:04,569 A ds times a dt, ds times a dt is right there. 341 00:18:04,569 --> 00:18:12,419 If we multiply this times the cross product of the partial 342 00:18:12,420 --> 00:18:17,740 derivative, this times this is going to give us this area. 343 00:18:17,740 --> 00:18:22,940 So if we summed up all of this times this, or this times this, 344 00:18:22,940 --> 00:18:26,490 if we summed them up over this entire region, we will get all 345 00:18:26,490 --> 00:18:28,210 of the parallelograms in this region. 346 00:18:28,210 --> 00:18:30,279 We will get the surface area. 347 00:18:30,279 --> 00:18:33,869 So we can write-- I know this is all a little bit convoluted. 348 00:18:33,869 --> 00:18:35,809 And you need to kind of ponder a little bit. 349 00:18:35,809 --> 00:18:37,829 Surface intervals, at least in my head, are one of the hardest 350 00:18:37,829 --> 00:18:40,909 things to really visualize, but it all hopefully makes sense. 351 00:18:40,910 --> 00:18:44,380 So we can say that this thing right over here, the sum of all 352 00:18:44,380 --> 00:18:49,400 of the little parallelograms on our surface, or the surface 353 00:18:49,400 --> 00:18:53,250 area, is going to be equal to, instead of taking the sum over 354 00:18:53,250 --> 00:18:58,279 the surface, let's take the sum of all the ds times dt's over 355 00:18:58,279 --> 00:19:00,430 this region right here. 356 00:19:00,430 --> 00:19:01,740 And of course, we're also going to have to take 357 00:19:01,740 --> 00:19:02,750 this cross product here. 358 00:19:02,750 --> 00:19:04,369 And we know how to do that. 359 00:19:04,369 --> 00:19:06,339 That's a double integral. 360 00:19:06,339 --> 00:19:10,439 for going to take the double integral over this, we could 361 00:19:10,440 --> 00:19:14,400 call it this region, or this area, right here. 362 00:19:14,400 --> 00:19:18,460 That area is the same thing as that whole area, right 363 00:19:18,460 --> 00:19:22,230 over there, of this thing. 364 00:19:22,230 --> 00:19:25,710 I'll just write it in yellow. 365 00:19:25,710 --> 00:19:30,079 Of the cross product of the partial of r with respect to 366 00:19:30,079 --> 00:19:35,549 s, and the partial of r with respect to t. 367 00:19:35,549 --> 00:19:38,099 ds and dt. 368 00:19:38,099 --> 00:19:41,599 And so you literally just take, and it seems very convoluted in 369 00:19:41,599 --> 00:19:43,799 how you're going to actually evaluate it, but we were able 370 00:19:43,799 --> 00:19:47,609 to express this thing called a surface-- well, this is a very 371 00:19:47,609 --> 00:19:49,709 simple surface integral-- in something that we can 372 00:19:49,710 --> 00:19:50,505 actually calculate. 373 00:19:50,505 --> 00:19:52,720 And in the next few videos, I'm going to show you examples 374 00:19:52,720 --> 00:19:54,450 of actually calculating it. 375 00:19:54,450 --> 00:19:57,080 Now, this right here will only give you the surface area. 376 00:19:57,079 --> 00:20:00,559 But what if, at every point here-- so over here what 377 00:20:00,559 --> 00:20:03,970 we've done in both of these expressions, is we're just 378 00:20:03,970 --> 00:20:06,779 figuring out the surface area of each of these parallelograms 379 00:20:06,779 --> 00:20:08,379 and then adding them all up. 380 00:20:08,380 --> 00:20:09,430 That's what we're doing. 381 00:20:09,430 --> 00:20:12,480 But what if, associated with each of those little 382 00:20:12,480 --> 00:20:16,329 parallelograms, we had some value, where that value is 383 00:20:16,329 --> 00:20:24,029 defined by some third function f of x, y, z? 384 00:20:24,029 --> 00:20:27,240 So every parallelogram, it's super small, it's around a 385 00:20:27,240 --> 00:20:30,130 point, you can say it's maybe the center of it, doesn't 386 00:20:30,130 --> 00:20:30,880 have to be the center. 387 00:20:30,880 --> 00:20:33,350 But maybe the center of it is at some point in 388 00:20:33,349 --> 00:20:36,969 three-dimensional space, and if you use some other function, f 389 00:20:36,970 --> 00:20:40,079 of x, y, and z, you'll get the value of that point. 390 00:20:40,079 --> 00:20:44,779 And what we want to do is figure out what happens if 391 00:20:44,779 --> 00:20:47,609 for every one of those parallelograms, we were to 392 00:20:47,609 --> 00:20:51,250 multiply it times the value of the function at that point? 393 00:20:51,250 --> 00:20:53,450 So we could write it this way. 394 00:20:53,450 --> 00:20:55,230 So this is where, you can imagine, the function 395 00:20:55,230 --> 00:20:56,569 is just one. 396 00:20:56,569 --> 00:20:59,960 We're just multiplying each of the parallelograms by one. 397 00:20:59,960 --> 00:21:03,640 But we could imagine we're multiplying each of the little 398 00:21:03,640 --> 00:21:09,360 parallelograms by f of x, y, and z, d sigma, and it's going 399 00:21:09,359 --> 00:21:12,129 to be the exact same thing, where this is each of the 400 00:21:12,130 --> 00:21:14,080 little parallelograms, we're just going to multiply it 401 00:21:14,079 --> 00:21:16,509 by f of x, y, and z there. 402 00:21:16,509 --> 00:21:20,879 So we're going to integrate it over the area, over that 403 00:21:20,880 --> 00:21:27,120 region, of f of x, y, and z, and then times the magnitude of 404 00:21:27,119 --> 00:21:31,659 the partial of r with respect to f, crossed with the partial 405 00:21:31,660 --> 00:21:36,900 of r with respect to t, ds, dt. 406 00:21:36,900 --> 00:21:38,390 And of course, we're integrating with 407 00:21:38,390 --> 00:21:39,340 respect to s and t. 408 00:21:39,339 --> 00:21:42,119 Hopefully we can express this function in terms of s and 409 00:21:42,119 --> 00:21:43,589 t, and we should be able to, because we have a 410 00:21:43,589 --> 00:21:44,980 parameterization there. 411 00:21:44,980 --> 00:21:47,089 Wherever we see an x there, it's really x is 412 00:21:47,089 --> 00:21:48,509 a function of s and t. 413 00:21:48,509 --> 00:21:50,509 y is a function of s and t. 414 00:21:50,509 --> 00:21:52,529 z is a function of s and t. 415 00:21:52,529 --> 00:21:55,039 And this might look super convoluted and hard. 416 00:21:55,039 --> 00:21:57,579 And the visualizations for this, of why you'd want 417 00:21:57,579 --> 00:21:59,409 to do this, it has applications in physics. 418 00:21:59,410 --> 00:22:00,509 It's a little hard to visualize. 419 00:22:00,509 --> 00:22:04,779 It's easier just to visualize the straight-up surface area. 420 00:22:04,779 --> 00:22:07,139 But we're going to see the next few videos that it's a little 421 00:22:07,140 --> 00:22:09,650 hairy to calculate these problems, but they're 422 00:22:09,650 --> 00:22:10,740 not too hard to do. 423 00:22:10,740 --> 00:22:13,950 That you just kind of have to stick with them.