1 00:00:00,385 --> 00:00:02,515 Now that we hopefully have a conceptual understanding... 2 00:00:02,515 --> 00:00:05,350 ...of what a surface integral like this COULD represent, 3 00:00:05,350 --> 00:00:08,073 ...I want to think about how we can actually construct... 4 00:00:08,073 --> 00:00:10,009 ...a unit vector... 5 00:00:10,009 --> 00:00:13,669 ...a unit normal vector, at any point on the surface. 6 00:00:13,669 --> 00:00:15,483 And to do that, I will assume... 7 00:00:15,483 --> 00:00:18,221 ...that our surface can be parametrized... 8 00:00:18,221 --> 00:00:21,282 ...by the position vector function, r... 9 00:00:21,282 --> 00:00:23,432 ...and r is a function of two parameters. 10 00:00:23,432 --> 00:00:27,569 It's a function of u, and it is a function of v. 11 00:00:27,569 --> 00:00:29,529 You give me a u and a v and... 12 00:00:29,529 --> 00:00:31,489 ...that will essentially specify... 13 00:00:31,489 --> 00:00:35,002 ...a point on this two-dimensional surface right over here. 14 00:00:35,002 --> 00:00:37,931 It could be bent, so it kind of exists in three-dimensional space... 15 00:00:37,931 --> 00:00:42,738 But a u and a v will specify a given point on this surface. 16 00:00:42,969 --> 00:00:46,865 Now, let's think about what the directions of r... 17 00:00:46,865 --> 00:00:48,983 ...the partial of r with respect to... 18 00:00:49,399 --> 00:00:52,188 ...the partial of r with respect to u looks like... 19 00:00:52,188 --> 00:00:54,524 ...and what the direction of the partial of r... 20 00:00:54,524 --> 00:00:57,692 ...the partial of r with respect to v looks like. 21 00:00:57,953 --> 00:00:59,989 So let's say that we're at some... 22 00:00:59,989 --> 00:01:01,289 ...we're at some point. 23 00:01:01,828 --> 00:01:03,729 We're at some point, (u,v). 24 00:01:03,729 --> 00:01:06,635 So for some (u,v), if you'd find the position vector... 25 00:01:06,635 --> 00:01:09,457 ...it takes us to that point on the surface right over there. 26 00:01:09,457 --> 00:01:12,651 Now let's say that we increment u just a little bit. 27 00:01:12,651 --> 00:01:14,423 And as we increment u just a little bit, 28 00:01:14,423 --> 00:01:16,655 ...we're going to get to another point on our surface, 29 00:01:16,655 --> 00:01:18,417 ...and let's say that other point on the surface... 30 00:01:18,417 --> 00:01:19,461 ...is right over there. 31 00:01:19,630 --> 00:01:20,826 So what would r... 32 00:01:20,826 --> 00:01:23,121 What would this r_u vector look like? 33 00:01:23,121 --> 00:01:24,881 Well its magnitude is essentially going to be.. 34 00:01:24,881 --> 00:01:26,598 ...dependent on how fast it's happening, 35 00:01:26,598 --> 00:01:28,929 ...how fast we're moving towards that point, 36 00:01:28,929 --> 00:01:31,261 ...but its direction is going to be in that direction. 37 00:01:31,261 --> 00:01:32,806 It's going to be towards that point. 38 00:01:32,806 --> 00:01:34,251 It's going be along the surface. 39 00:01:34,251 --> 00:01:36,217 We're going from one point on the surface to another. 40 00:01:36,217 --> 00:01:38,760 It's essentially going to be tangent to the surface at that point. 41 00:01:38,760 --> 00:01:40,240 And I could draw a little bit bigger. 42 00:01:40,240 --> 00:01:42,181 It would look something like that. 43 00:01:42,181 --> 00:01:45,057 r... r_u. 44 00:01:45,057 --> 00:01:46,510 So I just zoomed in right over here. 45 00:01:47,110 --> 00:01:49,003 Now let's go back to this point. 46 00:01:49,003 --> 00:01:51,585 And now let's make v a little bit bigger. 47 00:01:51,585 --> 00:01:53,448 And let's say if we make v a little bit bigger, 48 00:01:53,448 --> 00:01:55,101 ...we go to this point right over here. 49 00:01:55,101 --> 00:01:58,625 So then our position vector, r, would point to this point. 50 00:01:58,625 --> 00:02:00,970 And so what would r_v look like? 51 00:02:00,970 --> 00:02:02,817 Well its magnitude, once again, would be dependent on... 52 00:02:02,817 --> 00:02:05,634 ...how fast we're going there, but the direction is what's interesting. 53 00:02:05,634 --> 00:02:09,078 The direction would also be tangential to the surface. 54 00:02:09,078 --> 00:02:11,090 We're going from one point on the surface to another... 55 00:02:11,090 --> 00:02:12,416 ...as we change v. 56 00:02:12,416 --> 00:02:15,356 So r_v might look something like that. 57 00:02:16,579 --> 00:02:17,853 And they're not necessarily... 58 00:02:17,853 --> 00:02:19,759 These two aren't necessarily perpendicular to each other. 59 00:02:19,759 --> 00:02:21,481 In fact, the way I drew them, they're not perpendicular. 60 00:02:21,835 --> 00:02:25,869 So r_v is like this, but they're both tangential to the plane. 61 00:02:25,869 --> 00:02:28,266 They're both essentially telling us, right at that point, 62 00:02:28,266 --> 00:02:31,139 ...what is the tangent? What is the slope in that... 63 00:02:31,139 --> 00:02:34,138 ...in the u direction, or in the v direction? 64 00:02:34,138 --> 00:02:35,720 Now, this is... 65 00:02:35,720 --> 00:02:37,067 When you have two... 66 00:02:37,067 --> 00:02:38,668 When you have two vectors that are... 67 00:02:38,668 --> 00:02:40,267 ...that are tangential to the plane, 68 00:02:40,267 --> 00:02:42,125 ...and they're not the same vector, these are actually... 69 00:02:42,125 --> 00:02:43,435 ...already specifying... 70 00:02:43,435 --> 00:02:45,784 ...these are already kind of determining a plane. 71 00:02:45,784 --> 00:02:48,417 And so you can imagine a plane that looks something like this. 72 00:02:48,417 --> 00:02:50,868 If you took linear combinations of these two things, 73 00:02:50,868 --> 00:02:54,099 ...you would get a plane that both of these would lie on. 74 00:02:54,099 --> 00:02:57,437 Now, we've done this before, but I'll re-visit it. 75 00:02:57,437 --> 00:02:59,669 What happens when I take the cross product... 76 00:02:59,669 --> 00:03:02,684 ...of r_u and r_v? 77 00:03:02,684 --> 00:03:05,673 What happens when I take the cross product? 78 00:03:05,673 --> 00:03:08,573 Well first, this is going to give us another vector. 79 00:03:08,573 --> 00:03:10,384 It's going to give us a vector... 80 00:03:10,384 --> 00:03:15,082 ...a vector that is perpendicular to both... 81 00:03:15,421 --> 00:03:19,051 ...to r_u AND r_v. 82 00:03:19,051 --> 00:03:20,823 Or another way to think about it is... 83 00:03:20,823 --> 00:03:22,867 ...this plane, that these... 84 00:03:22,867 --> 00:03:23,716 ...when you take the cross product... 85 00:03:23,716 --> 00:03:26,485 ...this plane is essentially a tangential plane... 86 00:03:26,485 --> 00:03:27,462 ...to the surface. 87 00:03:27,462 --> 00:03:28,984 And if something is going to be perpendicular... 88 00:03:28,984 --> 00:03:30,452 ...to both of these characters, 89 00:03:30,452 --> 00:03:32,555 ...it's going to have to be normal to them... 90 00:03:32,555 --> 00:03:33,723 ...or, it's definitely going to be perpendicular... 91 00:03:33,723 --> 00:03:35,442 ...to both of them, but it's going to be normal... 92 00:03:35,442 --> 00:03:36,467 ...to this plane. 93 00:03:36,467 --> 00:03:37,905 Which is essentially going to be... 94 00:03:37,905 --> 00:03:40,871 ...perpendicular to the surface itself. 95 00:03:40,871 --> 00:03:42,154 So this right over here... 96 00:03:42,154 --> 00:03:44,347 ...is going to be A normal vector. 97 00:03:44,347 --> 00:03:45,774 This is... I'll write it... 98 00:03:45,774 --> 00:03:46,956 Well, let me just write it this way. 99 00:03:46,956 --> 00:03:48,622 This is A normal vector. 100 00:03:49,176 --> 00:03:50,816 I'm not saying THE unit normal... 101 00:03:50,816 --> 00:03:52,906 I'm not saying THE normal vector, 'cause you have... 102 00:03:52,906 --> 00:03:54,442 ...you could have different normal vectors of... 103 00:03:54,442 --> 00:03:55,691 ...different magnitudes. 104 00:03:55,691 --> 00:03:58,861 This is A normal vector, when you take the cross product. 105 00:03:58,861 --> 00:04:01,641 And we can even think about what direction it's pointing in. 106 00:04:01,641 --> 00:04:04,477 And so when you have "something" cross "something else"... 107 00:04:04,477 --> 00:04:06,417 ...the easiest way I remember how to do it is... 108 00:04:06,417 --> 00:04:07,995 ...you point your left thumb... 109 00:04:07,995 --> 00:04:09,724 Oh, sorry. You point your RIGHT thumb... 110 00:04:09,724 --> 00:04:11,439 ...in the direction of the first vector... 111 00:04:11,439 --> 00:04:13,870 So, in this case, r_u. 112 00:04:13,870 --> 00:04:14,839 So let me see if I can... 113 00:04:14,839 --> 00:04:15,709 ...if I can draw this. 114 00:04:15,709 --> 00:04:17,771 I'm literally looking at my hand and trying to draw it. 115 00:04:18,402 --> 00:04:20,443 So you put your right thumb... 116 00:04:20,443 --> 00:04:22,303 -- so this is a right-hand rule, essentially -- 117 00:04:22,303 --> 00:04:24,339 ...in the direction of the first vector... 118 00:04:24,339 --> 00:04:28,200 ...and then you put your index finger in the direction of... 119 00:04:28,200 --> 00:04:29,634 ...the second vector... 120 00:04:29,634 --> 00:04:31,102 ...right over here. 121 00:04:31,102 --> 00:04:32,305 So this is the second vector. 122 00:04:32,305 --> 00:04:34,442 So that's the direction of my index finger. 123 00:04:35,026 --> 00:04:37,968 So my index finger is going to look something like... 124 00:04:37,968 --> 00:04:39,504 ...that. 125 00:04:39,504 --> 00:04:40,702 And then you bend... 126 00:04:40,702 --> 00:04:43,164 ...you bend your middle finger inward... 127 00:04:43,164 --> 00:04:45,367 ...and that will tell you the direction of the cross product. 128 00:04:45,367 --> 00:04:48,054 So if I bend my middle finger inward, 129 00:04:48,054 --> 00:04:49,533 ...it will look something... 130 00:04:49,533 --> 00:04:52,134 ...it will look something like this. 131 00:04:52,134 --> 00:04:54,651 And then of course, my other two fingers are just going to be... 132 00:04:54,651 --> 00:04:56,787 ...folded in like that, and they're not really relevant. 133 00:04:56,787 --> 00:04:59,389 But my other two fingers and my hand looks like that. 134 00:04:59,389 --> 00:05:00,924 And so that tells us the direction. 135 00:05:00,924 --> 00:05:02,265 The direction is going to be like that. 136 00:05:02,265 --> 00:05:03,616 It's going to be upward-facing. 137 00:05:03,616 --> 00:05:05,635 That's important, because you have normal vectors. 138 00:05:05,635 --> 00:05:06,672 One could... 139 00:05:06,672 --> 00:05:09,208 Or there's two directions of "normalcy," I guess you could say. 140 00:05:09,208 --> 00:05:11,156 One is going out like that... 141 00:05:11,156 --> 00:05:12,182 ...outwards... 142 00:05:12,182 --> 00:05:12,954 ...or I guess... 143 00:05:12,954 --> 00:05:14,434 ...in the upward direction... 144 00:05:14,434 --> 00:05:15,903 ...one would be going downwards, 145 00:05:15,903 --> 00:05:17,886 ...or going -- I guess you could say -- into the surface. 146 00:05:17,886 --> 00:05:19,524 But the way I have set it up right now, 147 00:05:19,524 --> 00:05:20,695 ...this would be going outwards. 148 00:05:20,695 --> 00:05:22,415 It would be A... 149 00:05:22,415 --> 00:05:24,878 It would be A normal vector... 150 00:05:24,878 --> 00:05:25,901 ...to the surface. 151 00:05:25,901 --> 00:05:28,068 Now, in order to go from A normal vector... 152 00:05:28,068 --> 00:05:31,016 ...to the UNIT normal vector, 153 00:05:31,016 --> 00:05:32,490 ...we just have to normalize it. 154 00:05:32,490 --> 00:05:33,684 We just have to divide this... 155 00:05:33,684 --> 00:05:35,173 ...by its magnitude. 156 00:05:35,173 --> 00:05:37,151 So now we have our drumroll. 157 00:05:37,151 --> 00:05:38,235 The unit vector... 158 00:05:38,235 --> 00:05:39,837 And it's going to essentially be... 159 00:05:39,837 --> 00:05:41,685 It's going to be a function of u... 160 00:05:41,685 --> 00:05:42,422 It's going to be... 161 00:05:42,422 --> 00:05:44,510 ...a function of u and v. 162 00:05:44,510 --> 00:05:45,888 You give me a u or a v... 163 00:05:45,888 --> 00:05:46,600 ...and I'll give you a... 164 00:05:46,600 --> 00:05:48,099 ...that unit normal vector. 165 00:05:48,099 --> 00:05:50,090 It is going to be equal to... 166 00:05:50,090 --> 00:05:52,058 ...the partial of r... 167 00:05:52,058 --> 00:05:53,998 ...the partial of r with respect to u... 168 00:05:53,998 --> 00:05:55,400 ...crossed with... 169 00:05:55,400 --> 00:05:58,052 ...the partial of r with respect to v... 170 00:05:58,052 --> 00:05:58,580 That just... 171 00:05:58,580 --> 00:06:00,090 Now that gives us A normal vector, 172 00:06:00,090 --> 00:06:01,451 but it hasn't been normalized. 173 00:06:01,451 --> 00:06:03,054 So we want to divide... 174 00:06:03,054 --> 00:06:04,550 ...by the magnitude... 175 00:06:04,550 --> 00:06:06,785 We want to divide by the magnitude... 176 00:06:06,785 --> 00:06:08,168 ...of the exact same thing. 177 00:06:08,168 --> 00:06:11,307 r_u crossed with r_v. 178 00:06:14,707 --> 00:06:15,417 And we're done! 179 00:06:15,417 --> 00:06:17,361 We have constructed a unit normal vector. 180 00:06:17,361 --> 00:06:21,569 And in future videos, we'll actually do this with concrete examples.