1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:19,650 at ocw.mit.edu. 8 00:00:34,690 --> 00:00:35,750 PROFESSOR: Hi. 9 00:00:35,750 --> 00:00:39,970 Our lesson today involves a rather subtle difference 10 00:00:39,970 --> 00:00:44,400 between a curve and a coordinate system. 11 00:00:44,400 --> 00:00:49,580 In other words, given a curve, that curve has a certain shape, 12 00:00:49,580 --> 00:00:53,360 has a certain position, independently 13 00:00:53,360 --> 00:00:55,510 of where our coordinate axes are if we're 14 00:00:55,510 --> 00:00:57,750 using Cartesian coordinates, or whether we're 15 00:00:57,750 --> 00:01:00,780 using other coordinate systems, or what have you, 16 00:01:00,780 --> 00:01:04,129 but the equation of the curve may very well 17 00:01:04,129 --> 00:01:08,630 depend on what coordinate system we're using. 18 00:01:08,630 --> 00:01:11,520 The thing that we would like to do in this particular lecture 19 00:01:11,520 --> 00:01:14,884 is to hit a very important highlight that comes up-- well, 20 00:01:14,884 --> 00:01:17,050 you can motivate it from a purely mathematical point 21 00:01:17,050 --> 00:01:19,590 of view, but physically there's an even more 22 00:01:19,590 --> 00:01:21,230 natural interpretation. 23 00:01:21,230 --> 00:01:22,970 And basically what the thing hinges on 24 00:01:22,970 --> 00:01:25,520 is this: If you're given a curve in space-- 25 00:01:25,520 --> 00:01:27,310 we'll start with a curve in the plane, 26 00:01:27,310 --> 00:01:30,720 but it applies to curves in space as well. 27 00:01:30,720 --> 00:01:34,990 Given that curve in the plane, does 28 00:01:34,990 --> 00:01:38,150 that curve have certain properties regardless 29 00:01:38,150 --> 00:01:41,750 of whether you know what your coordinate system is or not? 30 00:01:41,750 --> 00:01:45,960 Or in still other words, can you measure the shape of the curve, 31 00:01:45,960 --> 00:01:48,540 can you measure the speed along the curve 32 00:01:48,540 --> 00:01:51,490 as a particle traverses it, if you had never 33 00:01:51,490 --> 00:01:54,490 heard of the x- and y-coordinate system? 34 00:01:54,490 --> 00:01:57,000 And what this leads to is a new system 35 00:01:57,000 --> 00:02:01,070 of vectors by which we study motion in space 36 00:02:01,070 --> 00:02:04,340 called tangential and normal vectors when 37 00:02:04,340 --> 00:02:05,730 we're dealing in the plane. 38 00:02:05,730 --> 00:02:08,060 And there's a third number called the binormal vector, 39 00:02:08,060 --> 00:02:10,479 which we'll talk about later when we deal 40 00:02:10,479 --> 00:02:12,270 in three-dimensional space. 41 00:02:12,270 --> 00:02:15,860 Any rate, just for brevity, I call this lecture "Tangential 42 00:02:15,860 --> 00:02:18,130 and Normal Vectors". 43 00:02:18,130 --> 00:02:19,990 And the idea is something like this. 44 00:02:19,990 --> 00:02:24,960 We're given a curve C. Now, given this particular curve C, 45 00:02:24,960 --> 00:02:27,550 it happens that we have a Cartesian coordinate system 46 00:02:27,550 --> 00:02:28,450 here. 47 00:02:28,450 --> 00:02:30,680 And it also happens that we prefer, 48 00:02:30,680 --> 00:02:32,710 at least as we've done things in the past, 49 00:02:32,710 --> 00:02:35,750 we write everything in terms of i and j components. 50 00:02:35,750 --> 00:02:39,060 Notice that if a person were restricted to his universe 51 00:02:39,060 --> 00:02:43,700 being the curve C, i and j have no basic meaning to him. 52 00:02:43,700 --> 00:02:45,740 What does have a basic meaning to him, 53 00:02:45,740 --> 00:02:49,590 as he's moving along this curve, I would imagine, would be what? 54 00:02:49,590 --> 00:02:52,392 What is his motion tangential to the curve? 55 00:02:52,392 --> 00:02:54,850 In other words, if you want to look at this from a calculus 56 00:02:54,850 --> 00:02:57,550 point of view, if this is a smooth curve, 57 00:02:57,550 --> 00:03:00,680 in a sufficiently small neighborhood of this point, 58 00:03:00,680 --> 00:03:04,410 you cannot distinguish between the curve and the tangent line. 59 00:03:04,410 --> 00:03:06,720 And consequently, one could interpret 60 00:03:06,720 --> 00:03:10,000 that at a given instance the motion was always 61 00:03:10,000 --> 00:03:13,120 along the straight line tangential to the curve. 62 00:03:13,120 --> 00:03:15,640 What this leads to is the notion of inventing 63 00:03:15,640 --> 00:03:21,417 what we call a unit tangent vector, which I'll call T. 64 00:03:21,417 --> 00:03:22,750 And what is that tangent vector? 65 00:03:22,750 --> 00:03:24,520 It's not a constant, mind you. 66 00:03:24,520 --> 00:03:27,340 It shifts with position as you move along the curve. 67 00:03:27,340 --> 00:03:29,690 What is constant is its magnitude. 68 00:03:29,690 --> 00:03:31,725 It has constant magnitude 1. 69 00:03:31,725 --> 00:03:33,350 I guess what I'm trying to say, in sort 70 00:03:33,350 --> 00:03:37,790 of a surrealistic or metamathematical way, 71 00:03:37,790 --> 00:03:42,460 is that T plays to a person who's living on the curve C 72 00:03:42,460 --> 00:03:45,830 the same role that i plays to a person living 73 00:03:45,830 --> 00:03:48,810 in our ordinary space, but somehow or other he sees T 74 00:03:48,810 --> 00:03:51,700 as a constant vector as he moves along the curve 75 00:03:51,700 --> 00:03:53,890 If he visualizes the curve as being a straight line. 76 00:03:53,890 --> 00:03:57,130 He always sees it tangential to his motion. 77 00:03:57,130 --> 00:04:03,440 Now, in the same way that j was a 90-degree positive rotation 78 00:04:03,440 --> 00:04:06,630 of i, one would like to mimic the i and j 79 00:04:06,630 --> 00:04:10,790 Cartesian coordinate system by inventing another unit 80 00:04:10,790 --> 00:04:13,190 vector, which is again what? 81 00:04:13,190 --> 00:04:18,110 A positive 90-degree rotation of T. 82 00:04:18,110 --> 00:04:20,880 And we'll call that vector N. 83 00:04:20,880 --> 00:04:25,500 So that now we have a new system of coordinates, T and N, 84 00:04:25,500 --> 00:04:27,910 new system of variables, or vectors, 85 00:04:27,910 --> 00:04:31,150 whereby we can now study motion along a curve in a very 86 00:04:31,150 --> 00:04:32,210 natural way. 87 00:04:32,210 --> 00:04:36,520 In other words, we talk about the unit tangent direction 88 00:04:36,520 --> 00:04:39,410 and the unit normal direction. 89 00:04:39,410 --> 00:04:41,035 And we have this thing now established. 90 00:04:43,950 --> 00:04:45,680 What we would like to do is to see 91 00:04:45,680 --> 00:04:49,000 what happens in our study of kinematics, motion 92 00:04:49,000 --> 00:04:51,950 in the plane, motion in space, if we work in terms 93 00:04:51,950 --> 00:04:54,500 of tangential and normal components now 94 00:04:54,500 --> 00:04:57,020 rather than in terms of i and j components. 95 00:04:57,020 --> 00:04:58,940 The first thing that we'd probably like to do 96 00:04:58,940 --> 00:05:01,400 is figure out how in the world do you compute T? 97 00:05:01,400 --> 00:05:05,400 Well, for example, let's take a particular application. 98 00:05:05,400 --> 00:05:08,740 Let's take the example that we were dealing with last time 99 00:05:08,740 --> 00:05:12,170 where we had the radius vector R-- in other the scalar 100 00:05:12,170 --> 00:05:14,540 function t, where t denoted time-- 101 00:05:14,540 --> 00:05:16,040 and we were dealing what? 102 00:05:16,040 --> 00:05:19,570 Motion in space, where the radius vector 103 00:05:19,570 --> 00:05:22,800 R was a function of t. 104 00:05:22,800 --> 00:05:25,040 And remember what we showed last time? 105 00:05:25,040 --> 00:05:31,480 We showed that the dR/dt, the velocity vector, has its what? 106 00:05:31,480 --> 00:05:33,760 Its direction is always tangent to the curve. 107 00:05:33,760 --> 00:05:35,410 We proved that last time. 108 00:05:35,410 --> 00:05:38,160 Well, as long as the dR/dt is always 109 00:05:38,160 --> 00:05:41,220 tangent to the curve, what prevents it from being a unit 110 00:05:41,220 --> 00:05:42,460 tangent vector? 111 00:05:42,460 --> 00:05:45,550 Well, nothing prevents it from being a tangent vector, 112 00:05:45,550 --> 00:05:47,440 because it's already tangent to the curve. 113 00:05:47,440 --> 00:05:50,960 All that could go wrong is that the magnitude of the dR/dt 114 00:05:50,960 --> 00:05:52,120 is not 1. 115 00:05:52,120 --> 00:05:52,750 Well, lookit. 116 00:05:52,750 --> 00:05:55,300 That's, again, a very simple point to fix up. 117 00:05:55,300 --> 00:05:58,690 Namely, if the magnitude of dR/dt is not 1, 118 00:05:58,690 --> 00:06:01,950 suppose we divide that vector by its magnitude. 119 00:06:01,950 --> 00:06:05,450 We have already seen that given any non-zero vector, if you 120 00:06:05,450 --> 00:06:07,730 divide that vector by its magnitude, 121 00:06:07,730 --> 00:06:12,524 you get the unit vector in the same direction as the vector 122 00:06:12,524 --> 00:06:13,440 that you started with. 123 00:06:13,440 --> 00:06:16,510 In other words, if I take the vector dR/dt, which 124 00:06:16,510 --> 00:06:19,440 is already tangential to the curve at the given point, 125 00:06:19,440 --> 00:06:22,480 and I divide that by the magnitude of the dR/dt, 126 00:06:22,480 --> 00:06:27,260 then I automatically get the unit tangent vector. 127 00:06:27,260 --> 00:06:28,642 Is that clear? 128 00:06:28,642 --> 00:06:31,500 Well, since nobody says no, I assume it is clear. 129 00:06:31,500 --> 00:06:32,090 Lookit. 130 00:06:32,090 --> 00:06:36,440 We also showed last time that the magnitude of dR/dt 131 00:06:36,440 --> 00:06:38,670 is speed along the curve. 132 00:06:38,670 --> 00:06:43,010 Speed along the curve happens to be called ds/dt. 133 00:06:43,010 --> 00:06:46,200 So, again, another name for the unit tangent vector 134 00:06:46,200 --> 00:06:51,640 is dR/dt divided by ds/dt. 135 00:06:51,640 --> 00:06:54,030 By the chain rule, we can cancel dt. 136 00:06:54,030 --> 00:06:56,280 And by the way, notice the chain rule 137 00:06:56,280 --> 00:06:59,110 applies for vector functions like this, the same 138 00:06:59,110 --> 00:07:01,140 as it did in part one of our course, 139 00:07:01,140 --> 00:07:03,360 by virtue of what we showed in the last unit-- 140 00:07:03,360 --> 00:07:07,350 namely, that every formula for derivatives that 141 00:07:07,350 --> 00:07:10,260 was true for scalar functions also 142 00:07:10,260 --> 00:07:12,420 happens to be true for what? 143 00:07:12,420 --> 00:07:15,130 Vector functions of a scalar variable. 144 00:07:15,130 --> 00:07:18,200 At any rate, notice then by the chain rule, another way 145 00:07:18,200 --> 00:07:22,390 of saying T is that it's the derivative of the position 146 00:07:22,390 --> 00:07:25,530 vector R with respect to the arc length s. 147 00:07:25,530 --> 00:07:27,260 I'd like to make one comment on this. 148 00:07:27,260 --> 00:07:29,100 It's important enough so that I will also 149 00:07:29,100 --> 00:07:31,860 make this comment in the notes as well when 150 00:07:31,860 --> 00:07:33,310 we're doing the exercises. 151 00:07:33,310 --> 00:07:35,420 The point is that in many textbooks, 152 00:07:35,420 --> 00:07:38,970 they will define T by saying it's dR/ds. 153 00:07:38,970 --> 00:07:43,200 Now, 999 times out of 1,000-- in fact, 154 00:07:43,200 --> 00:07:47,010 999 times out of 998 even-- you will never 155 00:07:47,010 --> 00:07:50,540 be given R as a function of arc length in the real world. 156 00:07:50,540 --> 00:07:52,510 In the real world, R is a function 157 00:07:52,510 --> 00:07:54,890 of some parameter, usually time. 158 00:07:54,890 --> 00:07:57,000 And the trouble that happens is if you 159 00:07:57,000 --> 00:07:59,470 try to use this definition, you find 160 00:07:59,470 --> 00:08:02,770 yourself trying to convert things into s, 161 00:08:02,770 --> 00:08:05,102 and this makes sort of a mess for you. 162 00:08:05,102 --> 00:08:07,310 The thing I would like to show you-- and, by the way, 163 00:08:07,310 --> 00:08:10,240 this does not depend on t standing for time. 164 00:08:10,240 --> 00:08:12,324 If t is any variable-- and we show 165 00:08:12,324 --> 00:08:13,740 this in the notes in the exercises 166 00:08:13,740 --> 00:08:17,530 again-- if t is any scalar, if you differentiate R 167 00:08:17,530 --> 00:08:20,800 with respect to that scalar, and divide that result 168 00:08:20,800 --> 00:08:24,100 by the magnitude of this vector, you wind up 169 00:08:24,100 --> 00:08:25,740 with the unit tangent vector. 170 00:08:25,740 --> 00:08:28,060 In other words, in a real-life problem, 171 00:08:28,060 --> 00:08:31,970 do not worry about converting R into a function of s. 172 00:08:31,970 --> 00:08:35,280 Simply differentiate R as it stands with respect 173 00:08:35,280 --> 00:08:38,150 to the given variable, divide by the magnitude 174 00:08:38,150 --> 00:08:40,850 of the derivative, and, presto, you 175 00:08:40,850 --> 00:08:42,990 have the unit tangent vector. 176 00:08:42,990 --> 00:08:44,970 Of course, you may ask, if it's so simple 177 00:08:44,970 --> 00:08:47,860 to do what I just said, why is it that every book defines 178 00:08:47,860 --> 00:08:48,940 it this way? 179 00:08:48,940 --> 00:08:50,410 The answer is rather interesting, 180 00:08:50,410 --> 00:08:52,660 and that is, we have just mentioned 181 00:08:52,660 --> 00:08:56,230 that we would like to believe that the shape of a curve 182 00:08:56,230 --> 00:08:58,140 depends only on the curve itself, 183 00:08:58,140 --> 00:09:00,480 not on how we parameterize it. 184 00:09:00,480 --> 00:09:02,680 The beauty of this particular definition 185 00:09:02,680 --> 00:09:09,610 simply says the natural parameter is arc length-- 186 00:09:09,610 --> 00:09:11,129 namely, arc length doesn't depend 187 00:09:11,129 --> 00:09:12,170 on any coordinate system. 188 00:09:12,170 --> 00:09:14,290 Given the curve, start at any point 189 00:09:14,290 --> 00:09:16,580 you want, and you can measure the arc length. 190 00:09:16,580 --> 00:09:19,720 So s is a very natural parameter that does not depend 191 00:09:19,720 --> 00:09:21,360 on the coordinate system. 192 00:09:21,360 --> 00:09:24,050 In other words, by defining the unit tangent vector 193 00:09:24,050 --> 00:09:29,890 to be dR/ds, you have a beautiful philosophically pure 194 00:09:29,890 --> 00:09:33,250 mathematical definition, because you have a definition which 195 00:09:33,250 --> 00:09:35,430 does not depend on any coordinate system 196 00:09:35,430 --> 00:09:37,200 or any unnatural parameter. 197 00:09:37,200 --> 00:09:40,010 But in practice, this is the way we compute the unit tangent 198 00:09:40,010 --> 00:09:40,943 vector. 199 00:09:40,943 --> 00:09:44,610 The question that comes up is how do you find the vector N? 200 00:09:44,610 --> 00:09:47,090 And I'm going to show you the traditional way of doing this 201 00:09:47,090 --> 00:09:51,210 before I jazz it up with a more modern approach. 202 00:09:51,210 --> 00:09:52,910 Let's look at T over here. 203 00:09:52,910 --> 00:09:56,010 Let's call phi the angle that the unit tangent vector 204 00:09:56,010 --> 00:09:59,520 makes with the curve here. 205 00:09:59,520 --> 00:10:02,210 Notice that in terms of this diagram, since the unit tangent 206 00:10:02,210 --> 00:10:06,740 vector has magnitude 1, the i component of it 207 00:10:06,740 --> 00:10:10,740 will be cosine phi and the j component will be sine phi. 208 00:10:10,740 --> 00:10:16,280 In other words, T is equal to cosine phi i plus sine phi j. 209 00:10:16,280 --> 00:10:17,630 Let's just differentiate. 210 00:10:17,630 --> 00:10:19,470 See, T is a function of phi here. 211 00:10:19,470 --> 00:10:22,430 Let's just take the derivative of T with respect to phi, 212 00:10:22,430 --> 00:10:24,340 and we get right away-- remember, now, 213 00:10:24,340 --> 00:10:26,590 we're getting the mileage out of this basic definition 214 00:10:26,590 --> 00:10:27,173 of derivative. 215 00:10:27,173 --> 00:10:28,960 That hasn't changed since last time. 216 00:10:28,960 --> 00:10:31,110 We just differentiate term by term here. 217 00:10:31,110 --> 00:10:35,460 We get minus sine phi i plus cosine phi j. 218 00:10:35,460 --> 00:10:40,510 Right away we observe that dT / d phi is still a unit vector. 219 00:10:40,510 --> 00:10:43,300 You see, it's components are minus sine phi and cosine phi, 220 00:10:43,300 --> 00:10:46,430 so its magnitude is still 1. 221 00:10:46,430 --> 00:10:49,150 And its slope is cosine phi over minus sine 222 00:10:49,150 --> 00:10:53,240 phi, which is the negative reciprocal of the slope here. 223 00:10:53,240 --> 00:10:54,820 In other words, what this shows us, 224 00:10:54,820 --> 00:10:56,280 through the traditional approach, 225 00:10:56,280 --> 00:10:59,010 is that whatever vector the dT / d phi is, 226 00:10:59,010 --> 00:11:02,450 it's a unit vector perpendicular to T. 227 00:11:02,450 --> 00:11:05,528 By the way, what that tells us right away is that dT / d 228 00:11:05,528 --> 00:11:07,930 phi must be either plus N or minus N 229 00:11:07,930 --> 00:11:09,260 before we go any further. 230 00:11:09,260 --> 00:11:10,420 Why? 231 00:11:10,420 --> 00:11:13,830 Because we already saw that positive N, the vector that we 232 00:11:13,830 --> 00:11:17,540 called N, was a positive 90-degree rotation of T. 233 00:11:17,540 --> 00:11:21,090 If we only knew that dT / d phi was 234 00:11:21,090 --> 00:11:23,550 a positive 90-degree rotation rather 235 00:11:23,550 --> 00:11:26,450 than a negative 90-degree rotation, we'd be home free. 236 00:11:26,450 --> 00:11:28,720 And, again, the beauty of trigonometry, 237 00:11:28,720 --> 00:11:31,590 in the non-surveyor's sense of the word, 238 00:11:31,590 --> 00:11:37,360 analytically is this-- that sort of having a premonition of what 239 00:11:37,360 --> 00:11:40,270 we'd like to be true, we simply verify 240 00:11:40,270 --> 00:11:43,190 the trigonometric identities that the cosine of phi 241 00:11:43,190 --> 00:11:46,390 plus 90 degrees is minus sine phi, 242 00:11:46,390 --> 00:11:48,420 and the sine of phi plus 90 degrees 243 00:11:48,420 --> 00:11:51,700 is cosine phi, so that dT / d phi is what? 244 00:11:51,700 --> 00:11:54,560 It's cosine phi plus 90 degrees i, 245 00:11:54,560 --> 00:11:57,340 plus sine phi plus 90 degrees j. 246 00:11:57,340 --> 00:12:00,640 And if we now compare this with this, 247 00:12:00,640 --> 00:12:03,990 we notice that we have exactly the same expression, 248 00:12:03,990 --> 00:12:08,170 except that the angle has been increased by a positive 90 249 00:12:08,170 --> 00:12:09,090 degrees. 250 00:12:09,090 --> 00:12:14,100 In other words, dT / d phi is a positive 90-degree rotation 251 00:12:14,100 --> 00:12:17,620 of T. Consequently, dT / d phi is the vector 252 00:12:17,620 --> 00:12:20,790 that was called N. OK? 253 00:12:20,790 --> 00:12:22,400 That's dT / d phi. 254 00:12:22,400 --> 00:12:24,780 Now, let's go back to our kinematics. 255 00:12:24,780 --> 00:12:26,490 We have T and N now. 256 00:12:26,490 --> 00:12:28,770 Let's talk about our velocity vector v, 257 00:12:28,770 --> 00:12:32,010 where R is still some function of time. 258 00:12:32,010 --> 00:12:34,330 By definition, v is dR/dt. 259 00:12:34,330 --> 00:12:35,780 That isn't going to change. 260 00:12:35,780 --> 00:12:37,650 V was dR/dt last time. 261 00:12:37,650 --> 00:12:40,100 It's going to be dR/dt this time. 262 00:12:40,100 --> 00:12:43,929 It's going to be dR/dt whenever we want to use it. 263 00:12:43,929 --> 00:12:45,970 The only difference is that instead of expressing 264 00:12:45,970 --> 00:12:48,265 this in terms of i and j, we now want to express it 265 00:12:48,265 --> 00:12:51,950 in terms of T and N. And notice that since v 266 00:12:51,950 --> 00:12:57,140 has as its direction the direction of the tangent line, 267 00:12:57,140 --> 00:13:01,220 and as its magnitude ds/dt-- we saw that last time-- notice 268 00:13:01,220 --> 00:13:04,970 that in terms of T, v is just a scalar multiple of the unit 269 00:13:04,970 --> 00:13:07,920 tangent vector T. And what scalar multiple is it? 270 00:13:07,920 --> 00:13:09,610 It's ds/dt. 271 00:13:09,610 --> 00:13:10,400 All right. 272 00:13:10,400 --> 00:13:11,430 All that says is what? 273 00:13:11,430 --> 00:13:14,550 That v is the vector in the direction of T 274 00:13:14,550 --> 00:13:18,500 whose magnitude is ds/dt, which is speed along the curve. 275 00:13:18,500 --> 00:13:20,350 I now want to find a. 276 00:13:20,350 --> 00:13:21,640 a is acceleration. 277 00:13:21,640 --> 00:13:24,280 It's the same acceleration that I was talking about before. 278 00:13:24,280 --> 00:13:26,100 It's dv/dt. 279 00:13:26,100 --> 00:13:28,460 The only thing that's going to change now 280 00:13:28,460 --> 00:13:31,150 is I am not going to change the acceleration vector. 281 00:13:31,150 --> 00:13:33,790 I am going to change how it looks, because now I'm 282 00:13:33,790 --> 00:13:38,000 going to try to find it in terms of T and N components. 283 00:13:38,000 --> 00:13:39,920 So what do I do here? 284 00:13:39,920 --> 00:13:44,440 Look at this expression for v. ds/dt is speed along the curve. 285 00:13:44,440 --> 00:13:47,420 That changes from time to time, in general. 286 00:13:47,420 --> 00:13:51,590 The unit tangent vector T is also a variable function of t, 287 00:13:51,590 --> 00:13:53,870 unless T happens to be a straight line 288 00:13:53,870 --> 00:13:57,025 through the origin-- namely, notice that the unit tangent 289 00:13:57,025 --> 00:13:59,650 vector, even though it always has unit length, 290 00:13:59,650 --> 00:14:03,520 changes its direction as we move along the curve. 291 00:14:03,520 --> 00:14:08,190 So in other words, both of these factors are functions of t. 292 00:14:08,190 --> 00:14:10,970 Consequently, to differentiate this with respect to t, 293 00:14:10,970 --> 00:14:12,820 we must use the product rule. 294 00:14:12,820 --> 00:14:15,490 And the fact that all of our differentiation formulas 295 00:14:15,490 --> 00:14:18,900 are true for vector and scalar combinations 296 00:14:18,900 --> 00:14:21,610 as well as for scalars, I now use the regular product 297 00:14:21,610 --> 00:14:23,340 rule-- namely, it's the derivative 298 00:14:23,340 --> 00:14:26,750 of the first factor times the second, 299 00:14:26,750 --> 00:14:31,200 plus the first factor times the derivative of the second. 300 00:14:31,200 --> 00:14:36,650 And I now have a expressed in terms of two vectors, T 301 00:14:36,650 --> 00:14:40,370 and the derivative of the unit vector T with respect 302 00:14:40,370 --> 00:14:41,420 to time t. 303 00:14:41,420 --> 00:14:45,300 And somehow or other, all that's wrong here 304 00:14:45,300 --> 00:14:49,120 is I would like to get this thing expressed in terms of N. 305 00:14:49,120 --> 00:14:51,390 You see, when I'm working with T and N components, 306 00:14:51,390 --> 00:14:54,250 I want my answer to depend on T and N. 307 00:14:54,250 --> 00:14:56,710 Now, here's where I become very shrewd. 308 00:14:56,710 --> 00:14:58,580 And, by the way, this is an insight 309 00:14:58,580 --> 00:15:01,185 that, if you're going to pick it up at all, 310 00:15:01,185 --> 00:15:04,610 you're either born with it or you pick it up with experience. 311 00:15:04,610 --> 00:15:08,170 But you just have to work with these things. 312 00:15:08,170 --> 00:15:10,750 There are tricks, if you want-- I guess 313 00:15:10,750 --> 00:15:12,950 the novice calls them "tricks." 314 00:15:12,950 --> 00:15:16,920 The expert calls it "keen analytical insight." 315 00:15:16,920 --> 00:15:19,760 The point is I want to get an N out of this thing. 316 00:15:19,760 --> 00:15:23,760 I already know how to express N in terms of dT / d phi. 317 00:15:23,760 --> 00:15:25,880 In fact, N is dT / d phi. 318 00:15:25,880 --> 00:15:29,560 So what I do is I take dT/dt and say, 319 00:15:29,560 --> 00:15:33,780 let me write it so I can get a dT / d phi factor out of this. 320 00:15:33,780 --> 00:15:37,540 I also want everything to be in terms of arc length 321 00:15:37,540 --> 00:15:39,830 so I can ultimately have an answer which doesn't 322 00:15:39,830 --> 00:15:41,850 depend on a coordinate system. 323 00:15:41,850 --> 00:15:45,360 So what I really do is I use the chain rule 324 00:15:45,360 --> 00:15:49,630 to express this factor in terms of what?-- these three factors. 325 00:15:49,630 --> 00:15:51,250 You see, according to the chain rule, 326 00:15:51,250 --> 00:15:53,710 the d phi here cancels the d phi here, 327 00:15:53,710 --> 00:15:56,710 the ds here cancels the ds here, and all I'm saying 328 00:15:56,710 --> 00:16:01,770 is that dT/dt can be written as dT / d phi times d phi / ds 329 00:16:01,770 --> 00:16:03,340 times ds/dt. 330 00:16:03,340 --> 00:16:05,410 Now we're in very good shape, you see. 331 00:16:05,410 --> 00:16:12,730 dT / d phi we already know is N. And ds/dt we already 332 00:16:12,730 --> 00:16:14,850 know can go with this. 333 00:16:14,850 --> 00:16:17,600 And the only new thing that we have to worry about 334 00:16:17,600 --> 00:16:19,080 is what is d phi / ds. 335 00:16:21,670 --> 00:16:25,350 See, again what so often happens, you apply logic, 336 00:16:25,350 --> 00:16:28,710 you get to a certain inescapable conclusion, 337 00:16:28,710 --> 00:16:31,030 and then if you have brand new terms, 338 00:16:31,030 --> 00:16:33,049 you have a choice between doing what?-- saying I 339 00:16:33,049 --> 00:16:35,340 don't like the new terms, I'm going to throw them away, 340 00:16:35,340 --> 00:16:37,700 or saying I like the result, I had better interpret 341 00:16:37,700 --> 00:16:39,110 what this new term means. 342 00:16:39,110 --> 00:16:41,650 All I want to show you is, is that the d phi / 343 00:16:41,650 --> 00:16:44,560 ds has a very natural interpretation-- 344 00:16:44,560 --> 00:16:46,755 namely, what is d phi / ds. 345 00:16:46,755 --> 00:16:48,380 Let me tell you what it's called first. 346 00:16:48,380 --> 00:16:50,560 It's usually denoted by the Greek letter kappa, 347 00:16:50,560 --> 00:16:52,470 and it's called curvature. 348 00:16:52,470 --> 00:16:56,990 Its reciprocal, 1 over kappa, is usually denoted by rho, 349 00:16:56,990 --> 00:16:59,140 and it's called the radius of curvature. 350 00:16:59,140 --> 00:17:01,260 And I give you plenty of drill on the stuff. 351 00:17:01,260 --> 00:17:03,360 I just want to mention what these words are now. 352 00:17:03,360 --> 00:17:05,500 In fact, part of the drill is that d 353 00:17:05,500 --> 00:17:08,630 phi / ds is not a very convenient thing to compute. 354 00:17:08,630 --> 00:17:11,130 Usually you're given y as some function of x. 355 00:17:11,130 --> 00:17:14,069 And many of the drill problems that we have in calculus 356 00:17:14,069 --> 00:17:17,630 ask questions like, how do you express d phi / ds in terms 357 00:17:17,630 --> 00:17:19,920 of y, dy, dx, et cetera? 358 00:17:19,920 --> 00:17:23,390 Those are problems that we can get into in more detail as we 359 00:17:23,390 --> 00:17:24,380 do the exercises. 360 00:17:24,380 --> 00:17:26,089 But all I wanted to do in this lecture 361 00:17:26,089 --> 00:17:29,940 is to show you why d phi / ds is such a natural thing. 362 00:17:29,940 --> 00:17:32,420 Look at the curve s. 363 00:17:32,420 --> 00:17:35,280 As you move along this curve, notice 364 00:17:35,280 --> 00:17:38,890 that the change in phi with respect to s in a way 365 00:17:38,890 --> 00:17:42,410 tells you how the shape of the curve is changing. 366 00:17:42,410 --> 00:17:46,200 In other words, d phi / ds measures 367 00:17:46,200 --> 00:17:50,330 how-- what could be a more natural word than curvature? 368 00:17:50,330 --> 00:17:53,490 See, as phi changes as you move along the curve, that's 369 00:17:53,490 --> 00:17:55,500 measuring how your curvature is changing. 370 00:17:55,500 --> 00:17:59,900 As an extreme case, notice if the curve were a straight line, 371 00:17:59,900 --> 00:18:04,760 d phi / ds would be 0, because phi would be a constant. 372 00:18:04,760 --> 00:18:08,560 d phi / ds would be 0, and the curvature of a straight line 373 00:18:08,560 --> 00:18:10,090 should be 0. 374 00:18:10,090 --> 00:18:14,260 At any rate, one defines d phi / ds to be the curvature. 375 00:18:14,260 --> 00:18:16,830 And, in fact, to play it safely, since s 376 00:18:16,830 --> 00:18:18,880 can have two different senses-- in other words, 377 00:18:18,880 --> 00:18:20,420 why couldn't somebody else say why don't you 378 00:18:20,420 --> 00:18:22,920 go this way along the curve, I don't know what the sense is? 379 00:18:22,920 --> 00:18:24,930 Usually what one does to play it safe 380 00:18:24,930 --> 00:18:27,970 is we put the absolute value signs around d phi / ds 381 00:18:27,970 --> 00:18:30,570 and just call the magnitude the curvature. 382 00:18:30,570 --> 00:18:32,480 And the punch line is that once I 383 00:18:32,480 --> 00:18:38,130 call d phi / ds the curvature, what I wind up with is what? 384 00:18:38,130 --> 00:18:41,780 Just substituting in here now, the acceleration vector is d^2 385 00:18:41,780 --> 00:18:47,410 s dt squared times T plus kappa ds/dt squared times N. 386 00:18:47,410 --> 00:18:51,170 By the way, this entire recipe is derived in the text. 387 00:18:51,170 --> 00:18:54,700 I have you do it again as a learning exercise because I 388 00:18:54,700 --> 00:18:56,200 want you to practice with this. 389 00:18:56,200 --> 00:18:59,400 And I make additional comments on this in the notes. 390 00:18:59,400 --> 00:19:02,920 The textbook makes additional comments on it 391 00:19:02,920 --> 00:19:05,577 in the text, which is where you'd expect it to be. 392 00:19:05,577 --> 00:19:07,160 And all I want you to see is that this 393 00:19:07,160 --> 00:19:09,326 is the same acceleration vector that we were talking 394 00:19:09,326 --> 00:19:11,470 about in the last lecture, only now 395 00:19:11,470 --> 00:19:13,250 we're talking about how it looks in terms 396 00:19:13,250 --> 00:19:15,390 of tangential and normal components instead 397 00:19:15,390 --> 00:19:17,040 of i and j components. 398 00:19:17,040 --> 00:19:17,540 OK? 399 00:19:17,540 --> 00:19:19,670 And what's so good about tangential and normal? 400 00:19:19,670 --> 00:19:21,810 What's so good about tangential and normal 401 00:19:21,810 --> 00:19:23,850 is that you're now moving along the curve 402 00:19:23,850 --> 00:19:26,290 rather than with respect to some isolated 403 00:19:26,290 --> 00:19:28,950 x- and y-coordinate system. 404 00:19:28,950 --> 00:19:32,400 By the way, in the last unit we showed a rather interesting 405 00:19:32,400 --> 00:19:38,420 result, that if T was any vector function of the scalar x, 406 00:19:38,420 --> 00:19:40,940 and the magnitude of T was a constant, 407 00:19:40,940 --> 00:19:44,670 then dT/dx was perpendicular to T. That was 408 00:19:44,670 --> 00:19:46,710 an exercise in the last unit. 409 00:19:46,710 --> 00:19:49,330 Now, the interesting point is that the modern approach 410 00:19:49,330 --> 00:19:51,780 to calculus says this-- why should 411 00:19:51,780 --> 00:19:54,010 we single out the xy-plane? 412 00:19:54,010 --> 00:19:57,250 After all, you can be given a particle moving through space, 413 00:19:57,250 --> 00:20:00,030 or you can be using a different coordinate system. 414 00:20:00,030 --> 00:20:02,910 The natural parameter is arc length. 415 00:20:02,910 --> 00:20:04,610 Consequently, the modern approach 416 00:20:04,610 --> 00:20:07,310 never talks about the angle phi or anything like this. 417 00:20:07,310 --> 00:20:11,220 The modern approach simply says this-- define the unit tangent 418 00:20:11,220 --> 00:20:12,060 vector as before. 419 00:20:15,260 --> 00:20:17,220 Because the magnitude of T is a constant, 420 00:20:17,220 --> 00:20:20,390 since dT/ds is already perpendicular to T, 421 00:20:20,390 --> 00:20:26,270 let's define a second vector N to be dT/ds divided 422 00:20:26,270 --> 00:20:27,570 by its magnitude. 423 00:20:27,570 --> 00:20:29,230 Again, the same old trick. 424 00:20:29,230 --> 00:20:30,290 What have we done here? 425 00:20:30,290 --> 00:20:33,900 We have simply taken dT/ds, which 426 00:20:33,900 --> 00:20:36,480 we know is perpendicular to T-- any scalar 427 00:20:36,480 --> 00:20:39,520 multiple of the dT/ds will still be perpendicular to T-- 428 00:20:39,520 --> 00:20:40,900 but now this is what? 429 00:20:40,900 --> 00:20:44,280 It's a unit vector because we've divided this vector 430 00:20:44,280 --> 00:20:45,570 by its magnitude. 431 00:20:45,570 --> 00:20:47,620 Therefore, N is a unit vector. 432 00:20:47,620 --> 00:20:48,510 And where is it? 433 00:20:48,510 --> 00:20:52,520 It's perpendicular to T. If we now cross-multiply, 434 00:20:52,520 --> 00:20:57,820 notice that dT/ds is equal to the magnitude of dT/ds times 435 00:20:57,820 --> 00:21:01,270 N. See, just cross-multiply. 436 00:21:01,270 --> 00:21:06,370 I now claim that the magnitude of dT/ds is just d. phi / ds. 437 00:21:06,370 --> 00:21:07,431 Now, why is that? 438 00:21:07,431 --> 00:21:09,180 I guess I should have planned this better, 439 00:21:09,180 --> 00:21:12,910 but let me come back to the previous board over here. 440 00:21:12,910 --> 00:21:16,560 Notice that since T is a constant vector, 441 00:21:16,560 --> 00:21:19,250 since T is a constant vector, how does it change? 442 00:21:19,250 --> 00:21:21,090 It can't change in magnitude because it 443 00:21:21,090 --> 00:21:22,530 has constant magnitude. 444 00:21:22,530 --> 00:21:26,080 Therefore, its only change must be due to direction alone. 445 00:21:26,080 --> 00:21:29,760 But the direction of T is measured by phi. 446 00:21:29,760 --> 00:21:32,357 In other words, if dT/ds is changing at all-- 447 00:21:32,357 --> 00:21:33,940 in other words, if this is a variable, 448 00:21:33,940 --> 00:21:36,190 it must be changing only in direction, 449 00:21:36,190 --> 00:21:39,610 because the magnitude of T is always 1. 450 00:21:39,610 --> 00:21:41,790 In other words, T cannot change in magnitude. 451 00:21:41,790 --> 00:21:44,110 It must therefore change only in direction. 452 00:21:44,110 --> 00:21:46,480 In other words, the magnitude of dT/ds 453 00:21:46,480 --> 00:21:49,100 is the same as the magnitude of d phi / ds. 454 00:21:49,100 --> 00:21:51,440 Recall that we just defined the magnitude of d phi / 455 00:21:51,440 --> 00:21:55,220 ds to be kappa, and therefore dT/ds 456 00:21:55,220 --> 00:22:00,210 is kappa N, the same way as in the traditional approach. 457 00:22:00,210 --> 00:22:03,200 The beauty of this approach is that we're no longer restricted 458 00:22:03,200 --> 00:22:05,010 to the xy-plane. 459 00:22:05,010 --> 00:22:06,570 We're not restricted to any plane. 460 00:22:06,570 --> 00:22:08,580 We're not restricted to any coordinate system. 461 00:22:08,580 --> 00:22:11,320 We can now, in fact, generalize this to go out 462 00:22:11,320 --> 00:22:12,620 into three dimensions. 463 00:22:12,620 --> 00:22:14,487 And, in fact, some of you will probably 464 00:22:14,487 --> 00:22:16,070 have enough difficulty with what we've 465 00:22:16,070 --> 00:22:19,340 done so far that you won't want to go into three dimensions. 466 00:22:19,340 --> 00:22:22,390 What I've done is I have made up an optional unit that 467 00:22:22,390 --> 00:22:25,150 follows this one, a unit which has no lecture. 468 00:22:25,150 --> 00:22:27,490 It simply has a batch of exercises 469 00:22:27,490 --> 00:22:30,284 for those who have mastered the material in this unit 470 00:22:30,284 --> 00:22:31,700 and would like to see what happens 471 00:22:31,700 --> 00:22:33,210 in three-dimensional space. 472 00:22:33,210 --> 00:22:35,200 And, after all, when you deal with real life 473 00:22:35,200 --> 00:22:37,450 orbit-type problems and things like this, 474 00:22:37,450 --> 00:22:39,080 notice that you do need the geometry 475 00:22:39,080 --> 00:22:40,950 of three-dimensional space for this. 476 00:22:40,950 --> 00:22:44,027 If you so desire, you can then do the optional unit. 477 00:22:44,027 --> 00:22:45,360 That's why it's called optional. 478 00:22:45,360 --> 00:22:46,526 You can skip it if you want. 479 00:22:46,526 --> 00:22:49,140 There's no loss of continuity if you should skip it, 480 00:22:49,140 --> 00:22:53,230 but in that optional unit I devote computational drill 481 00:22:53,230 --> 00:22:55,320 to what happens when our curve happens 482 00:22:55,320 --> 00:22:58,010 to be a three-dimensional space curve-- in other words, a curve 483 00:22:58,010 --> 00:22:59,520 that winds through space. 484 00:22:59,520 --> 00:23:03,390 Notice, by the way, that in the same way as before, 485 00:23:03,390 --> 00:23:09,940 I can write R of t is x of t i plus y of t j plus z of t k. 486 00:23:09,940 --> 00:23:13,120 where that is the vector form of the curve 487 00:23:13,120 --> 00:23:16,190 in Cartesian coordinates given in scalar form 488 00:23:16,190 --> 00:23:18,860 by the three equations, x is some function of t, 489 00:23:18,860 --> 00:23:22,810 y is some function of t, z is some function of t. 490 00:23:22,810 --> 00:23:26,295 Again, my claim is that if I just take the dR/ds, 491 00:23:26,295 --> 00:23:28,720 I still have the unit tangent vector. 492 00:23:28,720 --> 00:23:31,110 And to see that, just notice what we're saying here. 493 00:23:31,110 --> 00:23:32,680 This is a space curve now. 494 00:23:32,680 --> 00:23:34,560 I've taken a small segment of it. 495 00:23:34,560 --> 00:23:38,260 Here's R, here's R plus delta R, so this difference 496 00:23:38,260 --> 00:23:41,895 is delta R. Look what happens as you take delta R 497 00:23:41,895 --> 00:23:43,650 and divide it by delta s. 498 00:23:43,650 --> 00:23:46,070 First of all, the direction of delta R 499 00:23:46,070 --> 00:23:49,090 does become the direction of the tangent line 500 00:23:49,090 --> 00:23:51,450 as delta s approaches 0. 501 00:23:51,450 --> 00:23:54,340 So certainly we can believe that the direction of dR/ds 502 00:23:54,340 --> 00:23:57,060 is going to be the tangential direction. 503 00:23:57,060 --> 00:24:00,730 Also, if we invoke the result of geometry 504 00:24:00,730 --> 00:24:03,030 that we talked about in part one of our course, 505 00:24:03,030 --> 00:24:05,120 when we talked about sine theta over theta 506 00:24:05,120 --> 00:24:08,700 as theta approaches 0, the length of the arc 507 00:24:08,700 --> 00:24:14,000 is approximately the length of the chord for small segments. 508 00:24:14,000 --> 00:24:15,695 So, therefore, delta R over delta 509 00:24:15,695 --> 00:24:18,680 s in magnitude approaches 1. 510 00:24:18,680 --> 00:24:21,430 In other words, dR/ds is still the unit tangent vector, 511 00:24:21,430 --> 00:24:23,480 the same as before. 512 00:24:23,480 --> 00:24:25,810 Again from a computational point of view, 513 00:24:25,810 --> 00:24:29,950 to find dR/ds you do not rewrite this in terms of s. 514 00:24:29,950 --> 00:24:31,240 You simply do what? 515 00:24:31,240 --> 00:24:34,050 You take dR/dt, the same as before, 516 00:24:34,050 --> 00:24:38,680 divide by its magnitude, and you automatically have dR/ds. 517 00:24:38,680 --> 00:24:43,430 Similarly, once T is given, to find N, 518 00:24:43,430 --> 00:24:46,290 you simply differentiate T with respect to s 519 00:24:46,290 --> 00:24:47,860 and divide it by its magnitude. 520 00:24:47,860 --> 00:24:50,440 And again notice, even though I've written it again, 521 00:24:50,440 --> 00:24:52,920 if you look back to the first third of our blackboard, 522 00:24:52,920 --> 00:24:54,450 this is the same definition for N 523 00:24:54,450 --> 00:24:57,130 as before, because our original definition did not 524 00:24:57,130 --> 00:25:00,313 specify that the curve had to be in a particular plane. 525 00:25:00,313 --> 00:25:01,140 See? 526 00:25:01,140 --> 00:25:02,710 The same general definition. 527 00:25:02,710 --> 00:25:06,330 So I now have T and N. Now what do T and N do? 528 00:25:06,330 --> 00:25:09,190 T and N determine a plane. 529 00:25:09,190 --> 00:25:12,740 It's a plane which we call the osculating plane to the curve. 530 00:25:12,740 --> 00:25:15,120 That's the plane which sort of touches the curve 531 00:25:15,120 --> 00:25:16,490 at that particular moment. 532 00:25:16,490 --> 00:25:18,690 Remember, this curve is winding through space. 533 00:25:18,690 --> 00:25:22,300 And, again, this is done in more detail in the notes. 534 00:25:22,300 --> 00:25:24,720 Not quite as elegantly as going like this, 535 00:25:24,720 --> 00:25:26,840 but the idea is you have this plane that's 536 00:25:26,840 --> 00:25:28,920 shifting along with the curve. 537 00:25:28,920 --> 00:25:30,830 The only thing that's missing that causes 538 00:25:30,830 --> 00:25:34,240 new complications when you deal in three-dimensional space 539 00:25:34,240 --> 00:25:37,500 is that in the same way that T and N take the place of i 540 00:25:37,500 --> 00:25:39,910 and j in two-space, you need something 541 00:25:39,910 --> 00:25:42,990 that takes the place of k in three-dimensional space. 542 00:25:42,990 --> 00:25:45,170 What we do is-- again look at the structure-- 543 00:25:45,170 --> 00:25:49,940 we mimic how k is related to i and j and invent a new vector 544 00:25:49,940 --> 00:25:53,470 called the binormal, hence abbreviated B, 545 00:25:53,470 --> 00:25:57,230 which is simply defined to be T cross N, the vector that you 546 00:25:57,230 --> 00:26:01,690 get by rotating the unit vector T into the unit vector N 547 00:26:01,690 --> 00:26:08,220 through the smaller-- namely, the positive 90-degree-- angle. 548 00:26:08,220 --> 00:26:09,610 Now what is B? 549 00:26:09,610 --> 00:26:12,340 B is perpendicular to both T and N. In other words, 550 00:26:12,340 --> 00:26:16,410 B is a vector which is perpendicular to the osculating 551 00:26:16,410 --> 00:26:17,140 plane. 552 00:26:17,140 --> 00:26:20,817 Since B always has a constant magnitude, 553 00:26:20,817 --> 00:26:22,775 because T and N are always perpendicular-- see, 554 00:26:22,775 --> 00:26:25,400 B always has magnitude 1-- the point 555 00:26:25,400 --> 00:26:29,570 is that dB/ds, the magnitude of dB/ds, 556 00:26:29,570 --> 00:26:32,190 measures the twist of the curve. 557 00:26:32,190 --> 00:26:34,880 In other words, here's this tangent plane 558 00:26:34,880 --> 00:26:37,840 following a point, a particle, along the curve. 559 00:26:37,840 --> 00:26:39,560 And what you're saying is how fast 560 00:26:39,560 --> 00:26:42,230 the direction of that tangent plane is changing 561 00:26:42,230 --> 00:26:44,280 is measured by dB/ds. 562 00:26:44,280 --> 00:26:45,540 That is called the "twist." 563 00:26:45,540 --> 00:26:46,702 I call it the "twist." 564 00:26:46,702 --> 00:26:48,660 I put it in quotation marks because nobody else 565 00:26:48,660 --> 00:26:49,720 calls it the "twist." 566 00:26:49,720 --> 00:26:51,580 The formal name is the "torsion." 567 00:26:51,580 --> 00:26:52,080 See? 568 00:26:52,080 --> 00:26:53,205 This is called the torsion. 569 00:26:53,205 --> 00:26:55,060 I talk about that more in the notes. 570 00:26:55,060 --> 00:26:56,790 The point being, by the way, that 571 00:26:56,790 --> 00:27:00,000 notice that if dB/ds happens to be 0-- in other words, 572 00:27:00,000 --> 00:27:02,300 if B happens to be constant-- then 573 00:27:02,300 --> 00:27:03,930 the curve lies in the plane. 574 00:27:03,930 --> 00:27:05,580 We certainly recognize that. 575 00:27:05,580 --> 00:27:08,910 For example, if the curve happens to be in the xy-plane, 576 00:27:08,910 --> 00:27:13,240 notice that if T and N were i and j, 577 00:27:13,240 --> 00:27:17,180 i cross j would just be k, B would then be a constant. 578 00:27:17,180 --> 00:27:20,620 The derivative of a constant with respect to any variable 579 00:27:20,620 --> 00:27:24,030 is 0, and, therefore, when the curve does lie in the plane, 580 00:27:24,030 --> 00:27:26,310 the torsion, the twist, is 0. 581 00:27:26,310 --> 00:27:28,920 In other words, the torsion does for three-dimensional space 582 00:27:28,920 --> 00:27:32,430 what the curvature in a sense does for two-dimensional space. 583 00:27:32,430 --> 00:27:34,700 At any rate, our main aim is to get 584 00:27:34,700 --> 00:27:38,340 you familiar with some vector calculus, and if in doing this 585 00:27:38,340 --> 00:27:40,230 we can also help you learn how to use 586 00:27:40,230 --> 00:27:42,310 this stuff in some physical applications, 587 00:27:42,310 --> 00:27:44,700 that happens to be frosting on the cake. 588 00:27:44,700 --> 00:27:47,380 Next time, we are going to talk about the fact 589 00:27:47,380 --> 00:27:50,800 that we still have to invent additional coordinate systems, 590 00:27:50,800 --> 00:27:53,830 that i and j isn't enough, T and N isn't enough. 591 00:27:53,830 --> 00:27:56,730 Next time we're going to show why we need polar coordinates, 592 00:27:56,730 --> 00:27:58,660 but we'll worry about that next time. 593 00:27:58,660 --> 00:28:02,740 Until next time then, goodbye. 594 00:28:02,740 --> 00:28:05,110 Funding for the publication of this video 595 00:28:05,110 --> 00:28:09,990 was provided by the Gabriella and Paul Rosenbaum Foundation. 596 00:28:09,990 --> 00:28:14,160 Help OCW continue to provide free and open access to MIT 597 00:28:14,160 --> 00:28:21,870 courses by making a donation at ocw.mit.edu/donate.