1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:01,900 PROFESSOR: Plus 1. 3 00:00:01,900 --> 00:00:04,730 And next would be between-- this is where 4 00:00:04,730 --> 00:00:06,900 most people have the problem. 5 00:00:06,900 --> 00:00:09,000 They thought x is any real number. 6 00:00:09,000 --> 00:00:10,830 No-- no, no, no, no, no. 7 00:00:10,830 --> 00:00:12,420 You wanted a segment. 8 00:00:12,420 --> 00:00:15,750 x has the values between this value, 9 00:00:15,750 --> 00:00:18,190 whatever value's on this axis and that value. 10 00:00:18,190 --> 00:00:23,730 So x equals 1, x equals 2 are the end points. 11 00:00:23,730 --> 00:00:29,810 How do you write a parameterized equation? 12 00:00:29,810 --> 00:00:32,479 And that should help you very much on the web work 13 00:00:32,479 --> 00:00:38,337 homework on that problem for such a function. 14 00:00:38,337 --> 00:00:39,545 Well, you say, wait a minute. 15 00:00:39,545 --> 00:00:41,675 Magdalena, this is a linear function. 16 00:00:41,675 --> 00:00:42,610 It's a piece of cake. 17 00:00:42,610 --> 00:00:45,230 I have just x plus 1. 18 00:00:45,230 --> 00:00:47,690 I know how to deal with that. 19 00:00:47,690 --> 00:00:49,750 Yes, but I'm asking you something else. 20 00:00:49,750 --> 00:00:53,770 Rather than writing the explicit equation 21 00:00:53,770 --> 00:00:58,870 in Cartesian coordinates x and y, tell me what time it is. 22 00:00:58,870 --> 00:01:01,110 And then I'm going to travel in time. 23 00:01:01,110 --> 00:01:06,590 I want to travel in time, in space-time, on the segment, 24 00:01:06,590 --> 00:01:08,190 right? 25 00:01:08,190 --> 00:01:13,700 So why if x equals x plus 1 has what 26 00:01:13,700 --> 00:01:15,910 is that-- what parameterization has infinitely 27 00:01:15,910 --> 00:01:18,080 many parameterization? 28 00:01:18,080 --> 00:01:22,620 Somebody will say, ha, you told us that it has infinitely many. 29 00:01:22,620 --> 00:01:24,200 Why do you insist on one? 30 00:01:24,200 --> 00:01:27,765 Which one is the most natural and the easiest to grasp? 31 00:01:27,765 --> 00:01:30,425 STUDENT: Zero to one. 32 00:01:30,425 --> 00:01:32,790 PROFESSOR: Zero to one is not a parameterization. 33 00:01:32,790 --> 00:01:34,560 STUDENT: Times zero one. 34 00:01:34,560 --> 00:01:39,280 PROFESSOR: So, so, so what is the parametric equation 35 00:01:39,280 --> 00:01:41,190 of a curve in general? 36 00:01:41,190 --> 00:01:46,530 If I have a curve, y equals-- oh, I'll start with x. 37 00:01:46,530 --> 00:01:49,850 X equals x of t and y equals y of t 38 00:01:49,850 --> 00:01:56,700 represent the two parametric questions that 39 00:01:56,700 --> 00:02:00,080 give that curve's equation in plane-- 40 00:02:00,080 --> 00:02:03,620 in plane where the i of t belongs 41 00:02:03,620 --> 00:02:05,270 to a certain interval i. 42 00:02:05,270 --> 00:02:06,720 That's the mysterious interval. 43 00:02:06,720 --> 00:02:10,100 I don't really care about that in general. 44 00:02:10,100 --> 00:02:15,280 In my case, which one is the most natural parametrization, 45 00:02:15,280 --> 00:02:17,940 guys? 46 00:02:17,940 --> 00:02:19,690 Take x to be time. 47 00:02:19,690 --> 00:02:20,790 Say again, Magdalena. 48 00:02:20,790 --> 00:02:23,040 Take x to be time. 49 00:02:23,040 --> 00:02:26,870 And that will make your life easier. 50 00:02:26,870 --> 00:02:29,550 I take x to be time. 51 00:02:29,550 --> 00:02:33,460 And then y would be time plus 1. 52 00:02:33,460 --> 00:02:34,570 And I'm happy. 53 00:02:34,570 --> 00:02:39,380 So the way they asked you to enter your answer in web work 54 00:02:39,380 --> 00:02:45,050 was as r of t equals-- and it's blinking, blinking, 55 00:02:45,050 --> 00:02:46,885 interactive field for you. 56 00:02:46,885 --> 00:02:48,820 You say, OK, t? 57 00:02:48,820 --> 00:02:51,490 T what? 58 00:02:51,490 --> 00:02:53,660 And I'm not going to solve your problem. 59 00:02:53,660 --> 00:02:55,530 But your problem is similar. 60 00:02:55,530 --> 00:02:56,030 Why? 61 00:02:56,030 --> 00:03:04,020 Because r of t, which is the vector equation of your y 62 00:03:04,020 --> 00:03:09,670 or curve would give you the position vector, which is what? 63 00:03:09,670 --> 00:03:10,265 Wait a second. 64 00:03:10,265 --> 00:03:14,440 Let me finish. x of t times i plus y of t times 65 00:03:14,440 --> 00:03:17,320 j is the definition I gave last time. 66 00:03:17,320 --> 00:03:18,280 Go ahead. 67 00:03:18,280 --> 00:03:20,900 STUDENT: Where'd you get r of t and what is it? 68 00:03:20,900 --> 00:03:23,550 PROFESSOR: I already discussed it last time. 69 00:03:23,550 --> 00:03:27,300 So since I'm reviewing today, just 70 00:03:27,300 --> 00:03:30,020 reviewing today chapter 10, I really 71 00:03:30,020 --> 00:03:32,340 don't mind going over with you. 72 00:03:32,340 --> 00:03:34,780 But please keep in mind this is the first 73 00:03:34,780 --> 00:03:37,629 and the last time I'm going to review things 74 00:03:37,629 --> 00:03:38,420 with you last time. 75 00:03:38,420 --> 00:03:44,150 So what did you say a position vector is for a curve? 76 00:03:44,150 --> 00:03:46,953 When we talked about the drunken bug, 77 00:03:46,953 --> 00:03:50,720 we say the drunken bug is following a trajectory. 78 00:03:50,720 --> 00:03:53,870 He or she is struggling in time. 79 00:03:53,870 --> 00:04:00,620 I have a given frame xyz system of coordinates-- system 80 00:04:00,620 --> 00:04:03,650 of axes of coordinates with a certain origin. 81 00:04:03,650 --> 00:04:07,660 Thank God for this origin because you cannot refer 82 00:04:07,660 --> 00:04:11,070 to a position vector unless you have a frame-- 83 00:04:11,070 --> 00:04:14,190 an original frame, a position frame, initial frame. 84 00:04:14,190 --> 00:04:21,856 So r of t represents the vector that originates at the origin o 85 00:04:21,856 --> 00:04:28,480 and ends exactly at the position of your particle at time t. 86 00:04:28,480 --> 00:04:30,925 If you want, if you hate bugs, this 87 00:04:30,925 --> 00:04:35,370 is just the particle from physics that travels in time t. 88 00:04:35,370 --> 00:04:35,990 So-- 89 00:04:35,990 --> 00:04:39,731 STUDENT: OK, so the r of t is represented in the parent 90 00:04:39,731 --> 00:04:40,230 equation 91 00:04:40,230 --> 00:04:41,500 PROFESSOR: Yes, sir. 92 00:04:41,500 --> 00:04:42,530 Exactly. 93 00:04:42,530 --> 00:04:45,820 In a plane where z is 0-- so you imagine 94 00:04:45,820 --> 00:04:48,520 the z-axis coming at z0. 95 00:04:48,520 --> 00:04:50,520 This is the xy plane. 96 00:04:50,520 --> 00:04:52,960 And I'm very happy I have on the floor. 97 00:04:52,960 --> 00:04:54,570 This bug is on the floor. 98 00:04:54,570 --> 00:04:56,469 He doesn't want to know what's the dimension. 99 00:04:56,469 --> 00:04:57,510 So what's he going to do? 100 00:04:57,510 --> 00:05:02,480 He's going to say plus 0 times k that I don't care about 101 00:05:02,480 --> 00:05:05,919 because the position vector will be given by-- 102 00:05:05,919 --> 00:05:06,460 STUDENT: So-- 103 00:05:06,460 --> 00:05:07,876 PROFESSOR: --or for a plane curve. 104 00:05:07,876 --> 00:05:09,470 STUDENT: So if this was in 3D space 105 00:05:09,470 --> 00:05:14,180 and we had three equations so it was like z equals-- 106 00:05:14,180 --> 00:05:19,392 is equal to 2y plus x plus 1, then it would be-- then how 107 00:05:19,392 --> 00:05:20,605 would we do that? 108 00:05:20,605 --> 00:05:23,400 PROFESSOR: Let me remind us in general the way I pointed it 109 00:05:23,400 --> 00:05:24,780 out last. 110 00:05:24,780 --> 00:05:27,800 R of t in general as a position vector, 111 00:05:27,800 --> 00:05:29,540 we said many things about it. 112 00:05:29,540 --> 00:05:33,590 We said it is a smooth function. 113 00:05:33,590 --> 00:05:36,170 What does it mean differential role 114 00:05:36,170 --> 00:05:38,810 with derivative continuous? 115 00:05:38,810 --> 00:05:41,072 What did-- actually, that's c1. 116 00:05:41,072 --> 00:05:42,030 What else did they say? 117 00:05:42,030 --> 00:05:43,520 He said it's a regular. 118 00:05:43,520 --> 00:05:45,540 It's a regular vector function. 119 00:05:45,540 --> 00:05:46,700 What does it mean? 120 00:05:46,700 --> 00:05:49,040 It never stops, not even for a second. 121 00:05:49,040 --> 00:05:51,950 Well, the velocity of that is zero. 122 00:05:51,950 --> 00:05:53,850 When we introduced it-- all right, 123 00:05:53,850 --> 00:05:56,015 I cannot teach the whole thing all over again, 124 00:05:56,015 --> 00:05:59,990 but I'll be happy to do review just today. 125 00:05:59,990 --> 00:06:05,420 It's going to be x of ti plus y of tj plus z over k. 126 00:06:05,420 --> 00:06:07,220 That is a way to write it like that. 127 00:06:07,220 --> 00:06:13,220 Or the simpler way to write it as x of t, y of t, z of t. 128 00:06:13,220 --> 00:06:15,830 Now, if it involves using different notation, 129 00:06:15,830 --> 00:06:17,720 I want to warn you about that. 130 00:06:17,720 --> 00:06:21,700 Some people like to put braces like angular brackets. 131 00:06:21,700 --> 00:06:25,460 Or some people like because it's a vector. 132 00:06:25,460 --> 00:06:29,486 And that's the way they define vector Some people like just 133 00:06:29,486 --> 00:06:30,236 round parentheses. 134 00:06:30,236 --> 00:06:31,710 This is more practically. 135 00:06:31,710 --> 00:06:34,450 These are the coordinates of a position vector 136 00:06:34,450 --> 00:06:37,240 with respect to the ijk frame. 137 00:06:37,240 --> 00:06:40,420 So since we talked about this already, 138 00:06:40,420 --> 00:06:43,020 some simple examples have been given. 139 00:06:43,020 --> 00:06:45,160 One of them was a circling plane, 140 00:06:45,160 --> 00:06:48,070 another circling plane of a different speed, 141 00:06:48,070 --> 00:06:49,500 a segment of a line. 142 00:06:49,500 --> 00:06:50,970 This is the segment of a line. 143 00:06:50,970 --> 00:06:52,260 What else have we discussed? 144 00:06:52,260 --> 00:06:54,360 We discuss about something wilder, 145 00:06:54,360 --> 00:06:57,690 which was the helix at different speeds? 146 00:06:57,690 --> 00:07:01,190 All right, so very good question for him was-- so 147 00:07:01,190 --> 00:07:02,830 is this x of tt? 148 00:07:02,830 --> 00:07:03,380 Yes. 149 00:07:03,380 --> 00:07:05,420 Is this y of tt plus 1? 150 00:07:05,420 --> 00:07:05,920 Yes. 151 00:07:05,920 --> 00:07:08,760 Is this z of t 0 in my case? 152 00:07:08,760 --> 00:07:09,722 Precisely 153 00:07:09,722 --> 00:07:13,166 STUDENT: So if you gave value to z, 154 00:07:13,166 --> 00:07:16,610 what would you chose to make t parameterized? 155 00:07:16,610 --> 00:07:20,314 PROFESSOR: OK, t in general, if you are moving, 156 00:07:20,314 --> 00:07:22,480 you have an infinite motion that comes from nowhere, 157 00:07:22,480 --> 00:07:24,220 goes nowhere, right? 158 00:07:24,220 --> 00:07:28,770 OK, then you can say t is between minus 159 00:07:28,770 --> 00:07:29,920 infinity plus infinity. 160 00:07:29,920 --> 00:07:31,050 And that's your i-- 161 00:07:31,050 --> 00:07:32,300 STUDENT: But what I'm saying-- 162 00:07:32,300 --> 00:07:36,510 PROFESSOR: But-- but in your case-- in your case, 163 00:07:36,510 --> 00:07:40,370 you think oh, I know where I'm starting. 164 00:07:40,370 --> 00:07:44,230 So to that equals to 1, t must be 1. 165 00:07:44,230 --> 00:07:47,060 So I start my movement at 1 second 166 00:07:47,060 --> 00:07:52,690 and I end my movement at 2 seconds where x will be 2, 167 00:07:52,690 --> 00:07:54,580 and y will be 3. 168 00:07:54,580 --> 00:07:57,431 STUDENT: Well, I mean-- so you said x equals t. 169 00:07:57,431 --> 00:07:59,816 You took that from the y equals x plus 1. 170 00:07:59,816 --> 00:08:02,439 If you had the third variable t, what would you-- 171 00:08:02,439 --> 00:08:03,980 PROFESSOR: It's not a third variable. 172 00:08:03,980 --> 00:08:05,860 It's the time parameter. 173 00:08:05,860 --> 00:08:08,770 So I work in three variables-- xyz in space. 174 00:08:08,770 --> 00:08:10,810 Those are my space coordinates. 175 00:08:10,810 --> 00:08:14,090 The space coordinates are function of time. 176 00:08:14,090 --> 00:08:17,130 So it's all about physics. 177 00:08:17,130 --> 00:08:19,690 So mathematics sometimes becomes physics. 178 00:08:19,690 --> 00:08:22,945 Thank God we are sisters, even step-sisters. 179 00:08:22,945 --> 00:08:24,760 X is a function of t. 180 00:08:24,760 --> 00:08:26,102 Y is a function of t. 181 00:08:26,102 --> 00:08:28,430 Z is a function of t. 182 00:08:28,430 --> 00:08:29,030 Right? 183 00:08:29,030 --> 00:08:30,740 Am I answering your question or maybe 184 00:08:30,740 --> 00:08:33,010 I didn't quite understand the-- 185 00:08:33,010 --> 00:08:35,474 STUDENT: Well, I understand how to parameterize 186 00:08:35,474 --> 00:08:36,826 the idea of a plane. 187 00:08:36,826 --> 00:08:39,179 How do you do it in space though? 188 00:08:39,179 --> 00:08:42,220 PROFESSOR: In space-- in space, you're already here. 189 00:08:42,220 --> 00:08:46,370 So if you want to ride this not in plane but in space, 190 00:08:46,370 --> 00:08:51,380 your parametric equation is ti plus t plus 1j plus 0k, 191 00:08:51,380 --> 00:08:54,430 for this example, anywhere in r3. 192 00:08:54,430 --> 00:08:56,570 We live in r3. 193 00:08:56,570 --> 00:08:58,380 All righty? 194 00:08:58,380 --> 00:09:00,850 We live in r3. 195 00:09:00,850 --> 00:09:03,430 OK, let me give you more examples. 196 00:09:03,430 --> 00:09:05,920 Because I think I'm running out of time. 197 00:09:05,920 --> 00:09:09,170 But I still have to cover the material, 198 00:09:09,170 --> 00:09:11,120 eventually get somewhere. 199 00:09:11,120 --> 00:09:15,740 However, I want you to see more examples that will help 200 00:09:15,740 --> 00:09:18,610 you grasp this notion better. 201 00:09:18,610 --> 00:09:25,190 So guys, imagine that we have space r3-- that 202 00:09:25,190 --> 00:09:28,634 could be rn-- in which I have an origin 203 00:09:28,634 --> 00:09:31,586 and I have a [INAUDIBLE]. 204 00:09:31,586 --> 00:09:35,030 And somebody gives me a position vector 205 00:09:35,030 --> 00:09:38,500 for a motion that's a regular curve. 206 00:09:38,500 --> 00:09:44,760 And that's x of tri plus y is tj plus z of tk. 207 00:09:44,760 --> 00:09:49,180 And since his question is a very valid one, 208 00:09:49,180 --> 00:09:52,760 let's see what happens in a later case. 209 00:09:52,760 --> 00:09:56,410 So I'm going to deviate a little from my lesson plan. 210 00:09:56,410 --> 00:09:59,690 And I say let us be flexible and compare 211 00:09:59,690 --> 00:10:02,029 that with the inner curve. 212 00:10:02,029 --> 00:10:04,025 Because in the process of comparison, 213 00:10:04,025 --> 00:10:06,520 you learn a lot more. 214 00:10:06,520 --> 00:10:11,020 If I were to be right above my [INAUDIBLE] like that. 215 00:10:11,020 --> 00:10:17,771 So this is the spacial curve in our three imaginary trajectory 216 00:10:17,771 --> 00:10:20,206 run of a bug or a particle. 217 00:10:20,206 --> 00:10:24,030 As we said, this is the planar curve-- planar, 218 00:10:24,030 --> 00:10:28,706 parametrized curve in r2. 219 00:10:28,706 --> 00:10:29,550 What's different? 220 00:10:29,550 --> 00:10:31,390 What do we know about them? 221 00:10:31,390 --> 00:10:35,470 We clearly know section 10.2. 222 00:10:35,470 --> 00:10:38,900 What I hate in general about processors 223 00:10:38,900 --> 00:10:43,230 is if they are way too structured. 224 00:10:43,230 --> 00:10:47,220 Mathematics cannot be talking sections where you say, oh, 225 00:10:47,220 --> 00:10:51,786 section 10.2 is only about velocity and acceleration. 226 00:10:51,786 --> 00:10:55,130 But section 10.4 is about tangent unit vector 227 00:10:55,130 --> 00:10:56,740 and principle normal. 228 00:10:56,740 --> 00:10:58,840 Well, they are related. 229 00:10:58,840 --> 00:11:03,740 So it's only natural when we talk about section 10.2 230 00:11:03,740 --> 00:11:11,470 acceleration and velocity that from acceleration, you 231 00:11:11,470 --> 00:11:22,290 have a induced line to tangent unit vector-- tangent unit 232 00:11:22,290 --> 00:11:23,400 vector. 233 00:11:23,400 --> 00:11:28,450 And later on, you're going to compare acceleration 234 00:11:28,450 --> 00:11:30,610 with a normal principal vector. 235 00:11:30,610 --> 00:11:32,490 Sometimes, they are the same thing. 236 00:11:32,490 --> 00:11:35,020 Sometimes, they are not the same thing. 237 00:11:35,020 --> 00:11:38,560 It's a good idea to see when they are the same thing 238 00:11:38,560 --> 00:11:40,600 and when they are not. 239 00:11:40,600 --> 00:11:44,890 So in section 10.4, we will focus practically 240 00:11:44,890 --> 00:11:48,430 or t, n, and v, the Frenet frame and its consequences 241 00:11:48,430 --> 00:11:51,820 on curvature, we already talked about that a little bit. 242 00:11:51,820 --> 00:11:56,300 In 10.2, practically, we didn't cover much. 243 00:11:56,300 --> 00:11:59,380 I only told you about velocity, acceleration. 244 00:11:59,380 --> 00:12:03,150 However, I would like to review that for you. 245 00:12:03,150 --> 00:12:05,990 Because I don't want to risk losing you. 246 00:12:05,990 --> 00:12:07,900 I'm going to lose some of you anyway. 247 00:12:07,900 --> 00:12:10,120 Two people said this course is too hard for me. 248 00:12:10,120 --> 00:12:11,950 I'm going to drop. 249 00:12:11,950 --> 00:12:14,270 You are free to drop and I think it's better for you 250 00:12:14,270 --> 00:12:16,360 to drop than struggle. 251 00:12:16,360 --> 00:12:21,442 But as long as you can still learn and you can follow, 252 00:12:21,442 --> 00:12:22,660 you shouldn't drop. 253 00:12:22,660 --> 00:12:26,870 So try to see exactly how much you can handle. 254 00:12:26,870 --> 00:12:30,400 If you can handle just the regular section of calc three, 255 00:12:30,400 --> 00:12:32,150 go to that regular section. 256 00:12:32,150 --> 00:12:36,430 If you can handle more, if you are good at mathematics, 257 00:12:36,430 --> 00:12:39,089 if you have always been considered bright 258 00:12:39,089 --> 00:12:41,780 in mathematics in high school, let us stay here. 259 00:12:41,780 --> 00:12:43,030 Otherwise, go. 260 00:12:43,030 --> 00:12:44,280 Don't stay. 261 00:12:44,280 --> 00:12:48,790 All right, so the velocities are prime of t. 262 00:12:48,790 --> 00:12:51,680 The acceleration is our double prime of t. 263 00:12:51,680 --> 00:12:53,610 We have done that last time. 264 00:12:53,610 --> 00:12:55,330 We were very happy. 265 00:12:55,330 --> 00:12:58,430 What would happen in a planar curve seen on 2? 266 00:12:58,430 --> 00:13:02,460 The same thing, of course, except the last component 267 00:13:02,460 --> 00:13:03,800 is not there. 268 00:13:03,800 --> 00:13:06,820 It's part of ti plus y prime of tj. 269 00:13:06,820 --> 00:13:10,780 And there is a 0k in both cases. 270 00:13:10,780 --> 00:13:12,870 So all these are factors. 271 00:13:12,870 --> 00:13:15,295 At times, I'm not going to point that out anymore. 272 00:13:15,295 --> 00:13:18,000 273 00:13:18,000 --> 00:13:20,320 The derivation goes component-wise. 274 00:13:20,320 --> 00:13:24,890 So if you forgot how to derive or you want to drink and derive 275 00:13:24,890 --> 00:13:28,370 or something, then you don't belong in this class. 276 00:13:28,370 --> 00:13:32,250 So again, make sure you know the derivations and integrations 277 00:13:32,250 --> 00:13:33,779 really well. 278 00:13:33,779 --> 00:13:35,445 I'm going to work some examples out just 279 00:13:35,445 --> 00:13:36,720 to refresh your memory. 280 00:13:36,720 --> 00:13:40,450 But if you have struggled with differentiation and integration 281 00:13:40,450 --> 00:13:45,170 in Calc 1, then you do not do belong in this class. 282 00:13:45,170 --> 00:13:53,136 All right, let's see about speed. 283 00:13:53,136 --> 00:13:54,620 It's about speed. 284 00:13:54,620 --> 00:13:56,030 It's about time. 285 00:13:56,030 --> 00:14:00,090 It's about time to remember what the speed was. 286 00:14:00,090 --> 00:14:04,320 The speed was the absolute value or the magnitude. 287 00:14:04,320 --> 00:14:07,015 It's not an absolute value, but it's a magnitude 288 00:14:07,015 --> 00:14:08,590 of the velocity factor. 289 00:14:08,590 --> 00:14:11,100 This is the speed. 290 00:14:11,100 --> 00:14:13,552 And the same in this case. 291 00:14:13,552 --> 00:14:17,600 If I want to write an explicit formula because somebody 292 00:14:17,600 --> 00:14:21,130 asked me by email, can I write an explicit formula, of course. 293 00:14:21,130 --> 00:14:24,246 That's a piece of cake and you should know that from before. 294 00:14:24,246 --> 00:14:29,780 X prime of t squared plus y prime of t squared plus z 295 00:14:29,780 --> 00:14:34,110 prime of t squared under the square root. 296 00:14:34,110 --> 00:14:37,265 I was not going to insist on the planar curve. 297 00:14:37,265 --> 00:14:41,240 Of course the planar curve will have a speed that all of you 298 00:14:41,240 --> 00:14:42,420 know about. 299 00:14:42,420 --> 00:14:44,950 And that's going to be square root of x prime of t 300 00:14:44,950 --> 00:14:49,070 squared plus y root prime of t squared. 301 00:14:49,070 --> 00:14:53,340 You should do your own thinking to see what the particular case 302 00:14:53,340 --> 00:14:55,710 will become. 303 00:14:55,710 --> 00:14:58,430 However, I want to see if you understood 304 00:14:58,430 --> 00:15:01,780 what derives from that in the sense 305 00:15:01,780 --> 00:15:06,494 that you should know the length of a arc of a curve. 306 00:15:06,494 --> 00:15:09,890 What is the length of an arc of a curve? 307 00:15:09,890 --> 00:15:15,240 Well, we have to look back at Calculus 2 a little bit 308 00:15:15,240 --> 00:15:20,900 and remember that the length of an arc of a curve in Calculus 2 309 00:15:20,900 --> 00:15:24,390 was given by, what? 310 00:15:24,390 --> 00:15:30,050 So you say, well, yeah. 311 00:15:30,050 --> 00:15:31,410 That was a long time ago. 312 00:15:31,410 --> 00:15:33,400 Well, some of you already don't even 313 00:15:33,400 --> 00:15:39,650 remember that as being integral from a to b of square root of 1 314 00:15:39,650 --> 00:15:43,480 plus 1 prime of x squared dx. 315 00:15:43,480 --> 00:15:46,630 And you were freaking out thinking, oh my god, 316 00:15:46,630 --> 00:15:51,556 I don't see how this formula from Calc 2, 317 00:15:51,556 --> 00:15:55,170 the arc of a curve, had you travel between time 318 00:15:55,170 --> 00:16:01,740 equals a and time equals b will relate to this formula. 319 00:16:01,740 --> 00:16:03,440 So what happened in Calc 2? 320 00:16:03,440 --> 00:16:07,060 In Calc 2, hopefully, you have a good teacher. 321 00:16:07,060 --> 00:16:09,660 And hopefully, you've learned a lot. 322 00:16:09,660 --> 00:16:12,540 This is between a and b, right? 323 00:16:12,540 --> 00:16:14,210 What did they teach you in Calc 2? 324 00:16:14,210 --> 00:16:16,600 They taught you that you have to take 325 00:16:16,600 --> 00:16:18,600 integral from a to b of square root of 1 326 00:16:18,600 --> 00:16:21,020 plus y prime of x squared ds. 327 00:16:21,020 --> 00:16:21,810 Why? 328 00:16:21,810 --> 00:16:23,730 You never asked your teacher why. 329 00:16:23,730 --> 00:16:24,230 That's bad. 330 00:16:24,230 --> 00:16:25,922 You should do that. 331 00:16:25,922 --> 00:16:29,060 You should ask why every time. 332 00:16:29,060 --> 00:16:32,790 They make you swallow a formula via memorization 333 00:16:32,790 --> 00:16:35,472 without understanding this is the speed. 334 00:16:35,472 --> 00:16:37,710 And now I'm coming with the good news. 335 00:16:37,710 --> 00:16:39,990 I have a proof of that. 336 00:16:39,990 --> 00:16:42,320 I know what speed means when I'm moving 337 00:16:42,320 --> 00:16:46,810 along the arc of a curve in plane. 338 00:16:46,810 --> 00:16:51,145 OK, so what is the distance travelled between time equals A 339 00:16:51,145 --> 00:16:52,500 and time equals B? 340 00:16:52,500 --> 00:16:57,070 It's going to be integral form a to be of the speed, right? 341 00:16:57,070 --> 00:16:59,450 This is the same one I'm driving from-- level two-- 342 00:16:59,450 --> 00:17:01,711 Amarillo or anywhere else. 343 00:17:01,711 --> 00:17:02,210 There. 344 00:17:02,210 --> 00:17:05,069 Now, what they showed you and they fooled you 345 00:17:05,069 --> 00:17:10,618 into memorizing that is just a consequence of this formula 346 00:17:10,618 --> 00:17:12,530 because of what he said. 347 00:17:12,530 --> 00:17:13,480 Why? 348 00:17:13,480 --> 00:17:16,785 The most usual parameterization is 349 00:17:16,785 --> 00:17:22,680 going to be y of t equals t-- I'm sorry, x of t equals vxst 350 00:17:22,680 --> 00:17:25,910 and y of t equals y of t. 351 00:17:25,910 --> 00:17:27,940 So, again x is time. 352 00:17:27,940 --> 00:17:33,130 In many linear curves, you can take x to be time, thank God. 353 00:17:33,130 --> 00:17:38,560 And then your parametrization will be t comma y of t. 354 00:17:38,560 --> 00:17:40,640 Because x is t. 355 00:17:40,640 --> 00:17:43,150 And x prime of t will be 1. 356 00:17:43,150 --> 00:17:45,930 Y prime of t will be y prime of t. 357 00:17:45,930 --> 00:17:50,040 When you take them, squish them, square them, sum them up, 358 00:17:50,040 --> 00:17:51,990 you get exactly this one. 359 00:17:51,990 --> 00:17:54,402 But you notice this is the speed. 360 00:17:54,402 --> 00:17:56,070 What is this the speed? 361 00:17:56,070 --> 00:18:03,288 Of some value over prime of t, which is speed. 362 00:18:03,288 --> 00:18:07,250 You see that what they forced you to memorize in Calc 2 363 00:18:07,250 --> 00:18:10,920 is nothing but the speed. 364 00:18:10,920 --> 00:18:12,920 And I could change the parameterization 365 00:18:12,920 --> 00:18:14,980 to something more general. 366 00:18:14,980 --> 00:18:19,560 Now, can I do this parameterization for a circle? 367 00:18:19,560 --> 00:18:20,230 No. 368 00:18:20,230 --> 00:18:22,460 Why not? 369 00:18:22,460 --> 00:18:25,000 I could, but then I'd have to split 370 00:18:25,000 --> 00:18:26,760 into the upper part and lower part 371 00:18:26,760 --> 00:18:29,040 because the circle is not a graph. 372 00:18:29,040 --> 00:18:31,210 So I take t between this and that 373 00:18:31,210 --> 00:18:35,920 and then I have square root of 1 minus t squared on top. 374 00:18:35,920 --> 00:18:38,800 And underneath, I have minus square root of 1 375 00:18:38,800 --> 00:18:39,570 minus t squared. 376 00:18:39,570 --> 00:18:43,980 So I split the poor circle into a graph and another graph. 377 00:18:43,980 --> 00:18:45,230 And I do it separately. 378 00:18:45,230 --> 00:18:47,310 And I can still apply that. 379 00:18:47,310 --> 00:18:49,430 But only a fool would do that, right? 380 00:18:49,430 --> 00:18:52,900 So what does a smart mathematician do? 381 00:18:52,900 --> 00:18:54,670 A smart mathematician will say, OK, 382 00:18:54,670 --> 00:18:59,740 for the circle, x is cosine t, y is sine t. 383 00:18:59,740 --> 00:19:01,830 And that is the parameterization I'm 384 00:19:01,830 --> 00:19:04,170 going to use for this formula. 385 00:19:04,170 --> 00:19:05,900 And I get speed 1. 386 00:19:05,900 --> 00:19:08,620 And I'm going to be happy, right? 387 00:19:08,620 --> 00:19:10,970 So it's a lot easier to understand what 388 00:19:10,970 --> 00:19:13,280 a general parameterization is. 389 00:19:13,280 --> 00:19:19,490 What is the length of an arc of a curve for a curving space? 390 00:19:19,490 --> 00:19:20,910 There's the bug. 391 00:19:20,910 --> 00:19:22,110 Time equals t0. 392 00:19:22,110 --> 00:19:23,970 He's buzzing. 393 00:19:23,970 --> 00:19:26,180 And after 10 seconds, he will be at the end. 394 00:19:26,180 --> 00:19:30,620 So it goes, [BUZZING] jump. 395 00:19:30,620 --> 00:19:35,120 OK, how much did he travel? 396 00:19:35,120 --> 00:19:41,700 Integral from a to b of square root of x prime of t squared 397 00:19:41,700 --> 00:19:44,430 plus y prime of t squared plus z prime of t 398 00:19:44,430 --> 00:19:50,430 squared-- no matter what that position vector x of ty of t0 399 00:19:50,430 --> 00:19:51,360 give us. 400 00:19:51,360 --> 00:19:56,200 So you take the coordinates of the velocity vector. 401 00:19:56,200 --> 00:19:57,150 You look at them. 402 00:19:57,150 --> 00:19:57,900 You square them. 403 00:19:57,900 --> 00:19:59,280 You add them together. 404 00:19:59,280 --> 00:20:00,740 You put them under the square root. 405 00:20:00,740 --> 00:20:02,510 That's going to be the speed. 406 00:20:02,510 --> 00:20:06,370 And displacement is integral of speed. 407 00:20:06,370 --> 00:20:09,100 When you guys learned in school, your teacher 408 00:20:09,100 --> 00:20:10,935 oversimplified the things. 409 00:20:10,935 --> 00:20:12,950 What did your teacher say in physics? 410 00:20:12,950 --> 00:20:15,700 Space equals speed times time. 411 00:20:15,700 --> 00:20:16,680 Say it again. 412 00:20:16,680 --> 00:20:19,935 He said space traveled is speed times time. 413 00:20:19,935 --> 00:20:23,635 But he assumed the speed is constant or constant 414 00:20:23,635 --> 00:20:26,570 on portions-- like, speedswise constant. 415 00:20:26,570 --> 00:20:28,940 Well, if it's a constant, the speed 416 00:20:28,940 --> 00:20:30,790 will get the heck out of here. 417 00:20:30,790 --> 00:20:35,300 And then the space will be speed times b minus a. 418 00:20:35,300 --> 00:20:37,840 But b minus a is delta t. 419 00:20:37,840 --> 00:20:41,200 In mathematics, in physics, we say b minus a is delta t. 420 00:20:41,200 --> 00:20:44,720 That's the interval of time that the bug travels or the particle 421 00:20:44,720 --> 00:20:45,730 travels. 422 00:20:45,730 --> 00:20:47,870 So he or she was right. 423 00:20:47,870 --> 00:20:51,060 Space is speed times time, but it's not like 424 00:20:51,060 --> 00:20:53,670 that unless the speed is constant. 425 00:20:53,670 --> 00:20:55,830 So he oversimplified your knowledge 426 00:20:55,830 --> 00:20:57,520 of mathematics and physics. 427 00:20:57,520 --> 00:20:59,040 Now you see the truth. 428 00:20:59,040 --> 00:21:04,500 Space is integral of speed. 429 00:21:04,500 --> 00:21:06,340 OK, now we understand. 430 00:21:06,340 --> 00:21:09,830 And I promised you last time that after reviewing, 431 00:21:09,830 --> 00:21:13,730 I didn't even say I would review anything from 10.2 and 10.4. 432 00:21:13,730 --> 00:21:14,850 I promised you more. 433 00:21:14,850 --> 00:21:17,580 I promised you that I'm going to compute something that's 434 00:21:17,580 --> 00:21:23,960 out of 10.4 which is called a curvature of a helix 435 00:21:23,960 --> 00:21:25,230 in particular. 436 00:21:25,230 --> 00:21:29,680 Because we looked at curvature of a parametric curve 437 00:21:29,680 --> 00:21:31,190 in general. 438 00:21:31,190 --> 00:21:36,700 I want to organize the material of review from 10.2 and 10.4 439 00:21:36,700 --> 00:21:40,260 in a big problem just like you will have in the exams, 440 00:21:40,260 --> 00:21:42,347 in the midterm, and in the final. 441 00:21:42,347 --> 00:21:43,430 I don't want to scare you. 442 00:21:43,430 --> 00:21:45,920 I just want to prepare you better 443 00:21:45,920 --> 00:21:49,844 for the kind of multiple questions we are going to have. 444 00:21:49,844 --> 00:21:55,240 So let me give you a funny looking curve. 445 00:21:55,240 --> 00:21:59,430 I want you to think about it and tell me what it is. 446 00:21:59,430 --> 00:22:01,935 a and b are positive numbers. 447 00:22:01,935 --> 00:22:07,140 a cosine ba sine t bt will be some sort of funny trajectory. 448 00:22:07,140 --> 00:22:09,930 You are already familiar to that. 449 00:22:09,930 --> 00:22:13,414 Last time, I gave you an example where a was 4-- oh my god, 450 00:22:13,414 --> 00:22:14,550 I don't even remember. 451 00:22:14,550 --> 00:22:16,460 You'll need to help me. 452 00:22:16,460 --> 00:22:18,684 [INAUDIBLE] 453 00:22:18,684 --> 00:22:21,120 STUDENT: 4, 4, 3. 454 00:22:21,120 --> 00:22:24,760 PROFESSOR: I took those because they are Pythagorean numbers. 455 00:22:24,760 --> 00:22:26,360 So what does it mean? 456 00:22:26,360 --> 00:22:28,970 3 squared plus 4 squared equals 5 squared. 457 00:22:28,970 --> 00:22:31,920 I wanted the sum of them to be a perfect square. 458 00:22:31,920 --> 00:22:33,230 So I was playing games. 459 00:22:33,230 --> 00:22:36,590 You can do that for any a and b. 460 00:22:36,590 --> 00:22:37,710 Now, what do I want? 461 00:22:37,710 --> 00:22:43,620 A-- like in 10.2 where you write r prime of t, 462 00:22:43,620 --> 00:22:46,540 rewrite that double prime of t. 463 00:22:46,540 --> 00:22:49,730 So it's a complex problem. 464 00:22:49,730 --> 00:22:53,210 In b, I want you to find t and r prime 465 00:22:53,210 --> 00:22:55,750 of t over-- who remembers the formula? 466 00:22:55,750 --> 00:22:57,700 I shouldn't have spoon-fed you that. 467 00:22:57,700 --> 00:22:58,620 STUDENT: Absolute-- 468 00:22:58,620 --> 00:23:00,755 PROFESSOR: Absolute magnitude, actually. 469 00:23:00,755 --> 00:23:03,520 It's more correct to say magnitude, right? 470 00:23:03,520 --> 00:23:04,300 Very good. 471 00:23:04,300 --> 00:23:08,636 And what else did I spoon-feed you last name? 472 00:23:08,636 --> 00:23:10,280 I spoon-fed you n. 473 00:23:10,280 --> 00:23:13,970 Let's compute n as well as part of the problem 474 00:23:13,970 --> 00:23:21,200 t prime t over t prime of t magnitude. 475 00:23:21,200 --> 00:23:24,140 STUDENT: So you're looking for the tangent unit vector. 476 00:23:24,140 --> 00:23:25,497 PROFESSOR: Tangent unit vector? 477 00:23:25,497 --> 00:23:27,080 STUDENT: And then you're looking for-- 478 00:23:27,080 --> 00:23:27,913 PROFESSOR: Yes, sir. 479 00:23:27,913 --> 00:23:30,830 And-- OK, don't you like me to also give you 480 00:23:30,830 --> 00:23:34,050 something like a grading grid, how much everything 481 00:23:34,050 --> 00:23:35,320 would be worth. 482 00:23:35,320 --> 00:23:36,570 Imagine you're taking an exam. 483 00:23:36,570 --> 00:23:39,710 Why not put yourself in an exam mode 484 00:23:39,710 --> 00:23:44,130 so you don't freak out during the actual exam? 485 00:23:44,130 --> 00:23:47,680 C will be another question, something smart. 486 00:23:47,680 --> 00:24:02,480 Let's see-- reparameterize an arc length to a plane, a curve, 487 00:24:02,480 --> 00:24:05,380 rho of s. 488 00:24:05,380 --> 00:24:08,510 Why not r of s like some people call-- use it 489 00:24:08,510 --> 00:24:10,160 and some books use it? 490 00:24:10,160 --> 00:24:11,759 Because if you're reparameterizing s, 491 00:24:11,759 --> 00:24:13,467 it's going to be the same physical limits 492 00:24:13,467 --> 00:24:15,870 but a different function. 493 00:24:15,870 --> 00:24:19,640 So if you remember the diagram I wrote before, 494 00:24:19,640 --> 00:24:24,480 little r is a function that comes from integral i time 495 00:24:24,480 --> 00:24:29,450 integral 2r3 and rho would be coming from a j to r3. 496 00:24:29,450 --> 00:24:32,610 And what is the relationship between them? 497 00:24:32,610 --> 00:24:36,360 This is t goes to s and this is s goes to t. 498 00:24:36,360 --> 00:24:39,450 What is d I'm asking you? 499 00:24:39,450 --> 00:24:41,380 Well, if you're d and c, of course 500 00:24:41,380 --> 00:24:44,530 you know what the arc length parameter will be. 501 00:24:44,530 --> 00:24:49,630 It's going to be integral from 0 to t or any t0 here 502 00:24:49,630 --> 00:24:54,962 of the speed-- of the speed of the original function here 503 00:24:54,962 --> 00:24:56,310 of t. 504 00:24:56,310 --> 00:25:01,820 The tau-- maybe tau is better than the dummy variable t. 505 00:25:01,820 --> 00:25:05,242 And e I want. 506 00:25:05,242 --> 00:25:06,750 You say, how much more do you want? 507 00:25:06,750 --> 00:25:07,650 I want a lot. 508 00:25:07,650 --> 00:25:09,132 I'm a greedy person. 509 00:25:09,132 --> 00:25:13,840 I want the curvature of the curve. 510 00:25:13,840 --> 00:25:17,550 And you have to remind me. 511 00:25:17,550 --> 00:25:19,990 Some of you are very good students, better than me. 512 00:25:19,990 --> 00:25:23,622 I mean, I'm still behind with a research course 513 00:25:23,622 --> 00:25:25,080 that I have-- research paper i have 514 00:25:25,080 --> 00:25:29,786 to read in two days in biology. 515 00:25:29,786 --> 00:25:35,500 But this curvature of the curve had a very simple formula 516 00:25:35,500 --> 00:25:36,980 that we all love. 517 00:25:36,980 --> 00:25:40,120 For mathematicians, it's a piece of cake to remember it. 518 00:25:40,120 --> 00:25:43,310 K-- that's what I like about being a mathematician. 519 00:25:43,310 --> 00:25:45,350 I don't need a good memory. 520 00:25:45,350 --> 00:25:47,920 Now I remember why I didn't go to medical school-- 521 00:25:47,920 --> 00:25:51,010 because my father told me, well, you 522 00:25:51,010 --> 00:25:53,810 should be able to remember all the bones in a person's body. 523 00:25:53,810 --> 00:25:55,890 And I said, dad, do you know all these names? 524 00:25:55,890 --> 00:25:56,210 Yes, of course. 525 00:25:56,210 --> 00:25:57,293 And he started telling me. 526 00:25:57,293 --> 00:26:01,410 Well, I realized that I would never remember those. 527 00:26:01,410 --> 00:26:07,030 But I remember this formula which is r rho. 528 00:26:07,030 --> 00:26:10,330 In this case, if our r is Greek rho, 529 00:26:10,330 --> 00:26:13,090 it's got to be rho double prime of what? 530 00:26:13,090 --> 00:26:15,970 of S. Is this correct, what I wrote? 531 00:26:15,970 --> 00:26:16,470 No. 532 00:26:16,470 --> 00:26:18,050 What's missing? 533 00:26:18,050 --> 00:26:22,520 The acceleration and arc length but in magnitude because that's 534 00:26:22,520 --> 00:26:23,905 a vector, of course. 535 00:26:23,905 --> 00:26:26,870 This is the scalar function. 536 00:26:26,870 --> 00:26:28,660 Anything else you want, Magdalena? 537 00:26:28,660 --> 00:26:30,190 Oh, that's enough. 538 00:26:30,190 --> 00:26:34,060 All right, so I want to know everything 539 00:26:34,060 --> 00:26:38,260 that's possible to know about this curve from 10.2 and 10.4 540 00:26:38,260 --> 00:26:39,890 sections. 541 00:26:39,890 --> 00:26:41,840 10.3-- skip 10.5. 542 00:26:41,840 --> 00:26:44,330 Skip-- you're happy about it. 543 00:26:44,330 --> 00:26:44,940 Yes sir. 544 00:26:44,940 --> 00:26:48,426 STUDENT: For the parameter on v, is it a t? 545 00:26:48,426 --> 00:26:49,920 And what's the integral? 546 00:26:49,920 --> 00:26:51,010 What's on the bottom. 547 00:26:51,010 --> 00:26:54,230 PROFESSOR: Ah, that value erased when I wrote that one. 548 00:26:54,230 --> 00:26:56,200 It was there-- t0. 549 00:26:56,200 --> 00:27:00,510 So I can start with any fixed t0 as my initial moment in time. 550 00:27:00,510 --> 00:27:02,560 I would like my initial moment in time 551 00:27:02,560 --> 00:27:05,980 to be 0 just to make my things easier. 552 00:27:05,980 --> 00:27:07,940 Are we ready to solve this problem together? 553 00:27:07,940 --> 00:27:11,570 I think we have just about the exact time 554 00:27:11,570 --> 00:27:14,070 we need to do everything. 555 00:27:14,070 --> 00:27:17,610 First of all, you have to tell me what kind of curve this is. 556 00:27:17,610 --> 00:27:20,020 Of course you know because you were here last time. 557 00:27:20,020 --> 00:27:23,250 Don't skip classes because you are missing everything out 558 00:27:23,250 --> 00:27:25,380 and then you will have to drop or withdraw. 559 00:27:25,380 --> 00:27:27,230 So don't skip class. 560 00:27:27,230 --> 00:27:31,160 What was that from last time? 561 00:27:31,160 --> 00:27:33,510 It was a helix. 562 00:27:33,510 --> 00:27:35,280 I'm going to try and redraw it. 563 00:27:35,280 --> 00:27:38,010 I know I'm wasting my time, but I would 564 00:27:38,010 --> 00:27:43,750 try to draw a better curve. 565 00:27:43,750 --> 00:27:46,325 Ah, what's the equation of the cylinder? 566 00:27:46,325 --> 00:27:49,938 [CLASS MURMURS] 567 00:27:49,938 --> 00:27:51,327 PROFESSOR: Huh? 568 00:27:51,327 --> 00:27:53,382 What's the equation of the cylinder? 569 00:27:53,382 --> 00:27:55,610 That's a quadratic that you are all 570 00:27:55,610 --> 00:28:01,252 familiar with on which on my beautiful helix is sitting on. 571 00:28:01,252 --> 00:28:02,980 I taught you the trick last time. 572 00:28:02,980 --> 00:28:04,350 Don't forget it. 573 00:28:04,350 --> 00:28:10,100 STUDENT: a over 4 cosine of t squared plus 8 over 4 sine 574 00:28:10,100 --> 00:28:10,850 of t squared. 575 00:28:10,850 --> 00:28:13,850 576 00:28:13,850 --> 00:28:16,250 PROFESSOR: So we do that-- very good. 577 00:28:16,250 --> 00:28:19,040 X is going to be-- let me right that down. 578 00:28:19,040 --> 00:28:20,425 X is cosine. 579 00:28:20,425 --> 00:28:22,940 Y is a sine t. 580 00:28:22,940 --> 00:28:24,610 And that's exactly what you asked me. 581 00:28:24,610 --> 00:28:26,060 And z is bt. 582 00:28:26,060 --> 00:28:29,740 And then what I need to do is square these guys out 583 00:28:29,740 --> 00:28:31,565 as you said very well. 584 00:28:31,565 --> 00:28:33,196 I don't care about this 2z. 585 00:28:33,196 --> 00:28:34,750 He's not in the picture here. 586 00:28:34,750 --> 00:28:38,820 X squared plus y squared will be a squared, which means I better 587 00:28:38,820 --> 00:28:42,900 go ahead and draw a circle of radius a on the bottom 588 00:28:42,900 --> 00:28:44,920 and then build my-- oh my god, it 589 00:28:44,920 --> 00:28:49,820 looks horrible-- the cylinder based on that circle. 590 00:28:49,820 --> 00:28:51,050 Guys, it's now straight. 591 00:28:51,050 --> 00:28:51,780 I'm sorry. 592 00:28:51,780 --> 00:28:55,380 I mean, I can do better than that. 593 00:28:55,380 --> 00:28:58,710 OK, good. 594 00:28:58,710 --> 00:29:02,760 So I'm starting at what point? 595 00:29:02,760 --> 00:29:06,334 I'm starting at a0 0 time t equals 0. 596 00:29:06,334 --> 00:29:07,500 We discussed that last time. 597 00:29:07,500 --> 00:29:09,300 I'm not going to repeat. 598 00:29:09,300 --> 00:29:12,300 I'm starting here, and two of you 599 00:29:12,300 --> 00:29:14,090 told me that if t equals phi over two, 600 00:29:14,090 --> 00:29:17,550 I'm going to be here and so on and so forth. 601 00:29:17,550 --> 00:29:21,842 If I ask you one more thing for extra credit, what 602 00:29:21,842 --> 00:29:30,970 is the length of the trajectory traveled by the bug, whatever 603 00:29:30,970 --> 00:29:38,380 that is, between time t equals 0 and time t equals phi over 2. 604 00:29:38,380 --> 00:29:40,080 I'd say that's extra credit. 605 00:29:40,080 --> 00:29:52,400 So, oh my god, 20%, 20%, 20%, 20%, 20%, and 10% for this one. 606 00:29:52,400 --> 00:29:56,570 And if you think why does she care about the percentages 607 00:29:56,570 --> 00:29:59,030 and points, you will care and I care. 608 00:29:59,030 --> 00:30:02,700 Because I want you to see how you are going to be assessed. 609 00:30:02,700 --> 00:30:05,460 If you have no idea how you're going to assessed, 610 00:30:05,460 --> 00:30:08,750 then you're going to be happy and i will be unhappy. 611 00:30:08,750 --> 00:30:12,030 All right, so for 20% credit on this problem, 612 00:30:12,030 --> 00:30:15,540 we want to see r prime of t will be, r double prime of t 613 00:30:15,540 --> 00:30:16,040 will be. 614 00:30:16,040 --> 00:30:18,095 That's going to be a piece of cake. 615 00:30:18,095 --> 00:30:21,420 And of course, it's maybe the reward is too big for that, 616 00:30:21,420 --> 00:30:23,200 but that's life. 617 00:30:23,200 --> 00:30:31,670 Minus a sine t a equals time t and d, d as in infinity. 618 00:30:31,670 --> 00:30:34,320 So I have an infinite cylinder on which 619 00:30:34,320 --> 00:30:37,230 I draw an infinite helix coming from hell 620 00:30:37,230 --> 00:30:39,370 and going to paradise. 621 00:30:39,370 --> 00:30:44,220 So between minus infinity and plus infinity, there's a guy. 622 00:30:44,220 --> 00:30:47,790 I'm going to draw a beautiful infinite helix. 623 00:30:47,790 --> 00:30:50,460 And this is what I posted here. 624 00:30:50,460 --> 00:30:53,260 What's the acceleration of this helix? 625 00:30:53,260 --> 00:30:59,600 Minus a cosine t minus 5 sine t and 0. 626 00:30:59,600 --> 00:31:03,280 Question, quick question for you. 627 00:31:03,280 --> 00:31:06,840 Will-- you guys are fast. 628 00:31:06,840 --> 00:31:10,640 Maybe I shouldn't go ahead of myself. 629 00:31:10,640 --> 00:31:14,530 Nobody's asking me what the speed is right now. 630 00:31:14,530 --> 00:31:17,760 So why would I do something that's not on the final, right? 631 00:31:17,760 --> 00:31:19,980 So let's see. 632 00:31:19,980 --> 00:31:23,130 T, you will have to compute the speed when you get to here. 633 00:31:23,130 --> 00:31:26,150 But wait until we get there. 634 00:31:26,150 --> 00:31:27,420 What is mister t? 635 00:31:27,420 --> 00:31:29,560 Mister t will be the tangent vector. 636 00:31:29,560 --> 00:31:34,860 So the velocity is going like a crazy guy, long vector. 637 00:31:34,860 --> 00:31:39,160 The normal unit vector says, I'm the tangent unit vector. 638 00:31:39,160 --> 00:31:42,750 I'm always perpendicular to the direction. 639 00:31:42,750 --> 00:31:43,590 I'm of length 1. 640 00:31:43,590 --> 00:31:47,491 STUDENT: I thought the tangent was parallel to the direction. 641 00:31:47,491 --> 00:31:49,490 PROFESSOR: Yes, the direction of motion is this. 642 00:31:49,490 --> 00:31:51,040 Look at me. 643 00:31:51,040 --> 00:31:53,220 This is my direction of motion. 644 00:31:53,220 --> 00:31:54,077 And the tangent is-- 645 00:31:54,077 --> 00:31:55,160 STUDENT: You said it was-- 646 00:31:55,160 --> 00:31:57,030 PROFESSOR: --in the direction of motion. 647 00:31:57,030 --> 00:31:57,690 STUDENT: But you said it was perpendicular. 648 00:31:57,690 --> 00:31:59,130 PROFESSOR: I said perpendicular? 649 00:31:59,130 --> 00:32:03,030 Because I was thinking ahead of myself and n. 650 00:32:03,030 --> 00:32:04,250 And I apologize. 651 00:32:04,250 --> 00:32:06,255 So thank you for correcting me. 652 00:32:06,255 --> 00:32:08,342 So t is the tangent unit vector. 653 00:32:08,342 --> 00:32:12,770 654 00:32:12,770 --> 00:32:15,230 I'm going along the direction of motion. 655 00:32:15,230 --> 00:32:17,690 And it's going to be perpendicular to t. 656 00:32:17,690 --> 00:32:22,263 And that's the principal normal unit vector-- 657 00:32:22,263 --> 00:32:24,227 principal normal unit vector. 658 00:32:24,227 --> 00:32:26,630 And you're going to tell me what I'm having here. 659 00:32:26,630 --> 00:32:27,560 Because I don't know. 660 00:32:27,560 --> 00:32:30,970 661 00:32:30,970 --> 00:32:37,410 T is minus a sine t a equals sine t 662 00:32:37,410 --> 00:32:41,120 and v divided by the speed. 663 00:32:41,120 --> 00:32:43,630 That's why I was getting ahead of myself 664 00:32:43,630 --> 00:32:46,920 thinking about the speed that you'll need later on anyway. 665 00:32:46,920 --> 00:32:50,240 But you already need it here, right? 666 00:32:50,240 --> 00:32:55,180 Because the denominator of this expression will be the speed. 667 00:32:55,180 --> 00:32:58,290 Magnitude of r prime-- what is that? 668 00:32:58,290 --> 00:33:01,470 Piece of cake-- square root of the sum 669 00:33:01,470 --> 00:33:05,747 of the squares of square root of a squared plus b squared. 670 00:33:05,747 --> 00:33:06,330 Piece of cake. 671 00:33:06,330 --> 00:33:07,290 I love it. 672 00:33:07,290 --> 00:33:09,004 So what do I notice? 673 00:33:09,004 --> 00:33:11,830 That although I'm going on a funny curve which 674 00:33:11,830 --> 00:33:15,650 is a parametrized helix, I expect some-- maybe 675 00:33:15,650 --> 00:33:18,210 I expected something wild in terms of speed. 676 00:33:18,210 --> 00:33:19,500 Well, the speed is constant. 677 00:33:19,500 --> 00:33:26,850 STUDENT: [INAUDIBLE] the square root of negative a sine t 678 00:33:26,850 --> 00:33:27,350 squared-- 679 00:33:27,350 --> 00:33:29,570 PROFESSOR: And what are those? 680 00:33:29,570 --> 00:33:33,470 A squared sine squared plus c squared cosine squared plus b 681 00:33:33,470 --> 00:33:35,512 squared, right? 682 00:33:35,512 --> 00:33:37,220 And what sine squared plus cosine squared 683 00:33:37,220 --> 00:33:38,450 is 1 [INAUDIBLE]. 684 00:33:38,450 --> 00:33:41,160 So you get a squared plus b squared. 685 00:33:41,160 --> 00:33:46,485 Good-- now let's go on and do the n. 686 00:33:46,485 --> 00:33:53,020 The n will be t prime over magnitude of t prime. 687 00:33:53,020 --> 00:33:56,350 When you do t prime, you'll say, wait a minute. 688 00:33:56,350 --> 00:33:59,752 I have square root of a squared plus b squared on the bottom. 689 00:33:59,752 --> 00:34:04,930 On the top, I have minus equals sine t minus a sine t and 0. 690 00:34:04,930 --> 00:34:06,310 We have time to finish? 691 00:34:06,310 --> 00:34:07,036 I think. 692 00:34:07,036 --> 00:34:08,590 I hope so. 693 00:34:08,590 --> 00:34:18,110 Divided by-- divided by the magnitude of this fellow. 694 00:34:18,110 --> 00:34:20,560 I will say, oh, wait a minute. 695 00:34:20,560 --> 00:34:24,371 The magnitude of this fellow is simply the magnitude 696 00:34:24,371 --> 00:34:26,364 of this over this magnitude. 697 00:34:26,364 --> 00:34:29,860 698 00:34:29,860 --> 00:34:34,449 And we've seen last time this is the magnitude of this vector a, 699 00:34:34,449 --> 00:34:35,190 right? 700 00:34:35,190 --> 00:34:35,840 Good. 701 00:34:35,840 --> 00:34:39,049 Now, so the answer will be n is going to be a unit 702 00:34:39,049 --> 00:34:41,920 vector, very nice friend of yours, minus cosine t 703 00:34:41,920 --> 00:34:44,146 minus sine t0. 704 00:34:44,146 --> 00:34:49,520 Can you draw a conclusion about how I should draw this vector? 705 00:34:49,520 --> 00:34:51,609 You see the component in k is 0. 706 00:34:51,609 --> 00:34:55,610 So this vector cannot be like that-- 707 00:34:55,610 --> 00:34:57,721 cannot be inclined with respect to the horizontal. 708 00:34:57,721 --> 00:34:58,220 Yes sir. 709 00:34:58,220 --> 00:35:00,487 STUDENT: So what happens to-- down there-- square root 710 00:35:00,487 --> 00:35:01,854 of a squared plus b squared? 711 00:35:01,854 --> 00:35:02,895 PROFESSOR: They simplify. 712 00:35:02,895 --> 00:35:04,414 This is division. 713 00:35:04,414 --> 00:35:05,080 STUDENT: Oh, OK. 714 00:35:05,080 --> 00:35:07,730 PROFESSOR: So this simplifies with that and a simplifies 715 00:35:07,730 --> 00:35:10,000 with a. 716 00:35:10,000 --> 00:35:12,090 I should leave some things as an exercise, 717 00:35:12,090 --> 00:35:15,600 but this is an obvious one so I don't have to explain anything. 718 00:35:15,600 --> 00:35:18,950 Minus cosine t minus sine t-- if do 719 00:35:18,950 --> 00:35:22,290 you guys imagine what that is? 720 00:35:22,290 --> 00:35:25,760 Remember your washer and dryer. 721 00:35:25,760 --> 00:35:32,500 So if you have an acceleration that's pointing inside 722 00:35:32,500 --> 00:35:36,170 like from a centrifugal force, the corresponding acceleration 723 00:35:36,170 --> 00:35:39,480 would go pointing inside, not outside. 724 00:35:39,480 --> 00:35:43,780 That's going to be exactly minus cosine t minus sine t0. 725 00:35:43,780 --> 00:35:47,520 So the way I should draw the n would not be just any n, 726 00:35:47,520 --> 00:35:52,660 but should be at every point a beautiful vector 727 00:35:52,660 --> 00:35:55,770 that's horizontal and is moving along the helix. 728 00:35:55,770 --> 00:35:57,840 My elbow is moving along the helix. 729 00:35:57,840 --> 00:35:58,630 See my elbow? 730 00:35:58,630 --> 00:35:59,965 Where's my elbow moving? 731 00:35:59,965 --> 00:36:01,200 I'm trying. 732 00:36:01,200 --> 00:36:03,070 I swear, I won't do it that way. 733 00:36:03,070 --> 00:36:06,910 So this is the helix and this is the acceleration, which 734 00:36:06,910 --> 00:36:12,525 is acceleration and the normal unit vector in this case 735 00:36:12,525 --> 00:36:13,210 are co-linear. 736 00:36:13,210 --> 00:36:14,980 They are not co-linear in general. 737 00:36:14,980 --> 00:36:18,970 But if the speed is a constant, they are co-linear. 738 00:36:18,970 --> 00:36:20,816 The n and the acceleration. 739 00:36:20,816 --> 00:36:21,316 Yes, sir? 740 00:36:21,316 --> 00:36:24,774 STUDENT: How do you know it's pointing in the central axis? 741 00:36:24,774 --> 00:36:25,645 I thought it was-- 742 00:36:25,645 --> 00:36:26,686 PROFESSOR: Good question. 743 00:36:26,686 --> 00:36:27,956 Good question. 744 00:36:27,956 --> 00:36:28,890 Well, yeah. 745 00:36:28,890 --> 00:36:29,590 Let's see now. 746 00:36:29,590 --> 00:36:31,160 Plug in t equals 0. 747 00:36:31,160 --> 00:36:32,330 What do you have? 748 00:36:32,330 --> 00:36:35,820 Minus cosine 0 minus 1 0, 0. 749 00:36:35,820 --> 00:36:40,380 So you guys would have to draw the vector minus 1, 0, 0. 750 00:36:40,380 --> 00:36:42,310 That's minus i, right? 751 00:36:42,310 --> 00:36:47,980 So when I start here, this is my n-- from here to here, 752 00:36:47,980 --> 00:36:50,910 from the particle to the insid. 753 00:36:50,910 --> 00:36:52,720 So I go on that. 754 00:36:52,720 --> 00:36:55,100 All right, so this is the normal principal vector. 755 00:36:55,100 --> 00:36:57,066 I'm very happy about it. 756 00:36:57,066 --> 00:36:59,820 STUDENT: Isn't the normal principal vector is the-- is it 757 00:36:59,820 --> 00:37:01,440 the derivative of t, or is just-- 758 00:37:01,440 --> 00:37:02,815 PROFESSOR: It was by definition-- 759 00:37:02,815 --> 00:37:05,490 it's in your notes-- t prime over the magnitude of the-- 760 00:37:05,490 --> 00:37:09,100 STUDENT: So then did you-- why didn't you 761 00:37:09,100 --> 00:37:10,982 take a derivative of t prime? 762 00:37:10,982 --> 00:37:11,690 PROFESSOR: I did. 763 00:37:11,690 --> 00:37:12,606 STUDENT: Yeah, I know. 764 00:37:12,606 --> 00:37:15,880 I see you took a derivative of t of-- 765 00:37:15,880 --> 00:37:19,172 PROFESSOR: This is t prime. 766 00:37:19,172 --> 00:37:20,380 STUDENT: OK. 767 00:37:20,380 --> 00:37:23,570 PROFESSOR: And this is magnitude of t prime. 768 00:37:23,570 --> 00:37:26,360 Why don't you try this at home, like, 769 00:37:26,360 --> 00:37:30,020 slowly until you're sure this is what yo got? 770 00:37:30,020 --> 00:37:32,410 So I did-- I did the derivative of i. 771 00:37:32,410 --> 00:37:33,880 STUDENT: I saw that. 772 00:37:33,880 --> 00:37:36,060 PROFESSOR: This is a [INAUDIBLE]. 773 00:37:36,060 --> 00:37:37,950 STUDENT: You said you were-- 774 00:37:37,950 --> 00:37:40,340 PROFESSOR: So when we have t times a function 775 00:37:40,340 --> 00:37:43,460 and we prime the product, k goes out. 776 00:37:43,460 --> 00:37:45,380 Lucky for us-- imagine how life would 777 00:37:45,380 --> 00:37:46,970 be if it weren't like that. 778 00:37:46,970 --> 00:37:49,180 So the constant that falls out is 779 00:37:49,180 --> 00:37:51,890 1 over square root of what I derived. 780 00:37:51,890 --> 00:37:56,000 And then I have to derive this whole function also. 781 00:37:56,000 --> 00:37:59,050 So I would suggest to everybody, not just to yo-- 782 00:37:59,050 --> 00:38:01,751 go home and see if you can redo this 783 00:38:01,751 --> 00:38:03,000 without looking in your notes. 784 00:38:03,000 --> 00:38:05,100 Close the damn notes. 785 00:38:05,100 --> 00:38:08,780 Open and then you look at-- it's line by line, line by line 786 00:38:08,780 --> 00:38:10,620 all the derivations. 787 00:38:10,620 --> 00:38:13,970 Because you guys will have to do that yourselves in the exam, 788 00:38:13,970 --> 00:38:17,468 either midterm or final anyway. 789 00:38:17,468 --> 00:38:23,710 Reparameterizing arc lengths to obtain a curve-- I 790 00:38:23,710 --> 00:38:25,930 still have that to finish the problem. 791 00:38:25,930 --> 00:38:31,660 Reparameterizing arc length to obtain a curve rho of s. 792 00:38:31,660 --> 00:38:33,100 How do we do that? 793 00:38:33,100 --> 00:38:34,070 Who is s? 794 00:38:34,070 --> 00:38:37,300 First of all, you should start with the s and then 795 00:38:37,300 --> 00:38:38,660 reparameterize. 796 00:38:38,660 --> 00:38:39,820 So you say, hey, teacher. 797 00:38:39,820 --> 00:38:42,050 You try to fool me, right? 798 00:38:42,050 --> 00:38:45,590 I want s to be grabbed as a parameter first. 799 00:38:45,590 --> 00:38:49,700 And then I will reparameterize the way you want me to do that. 800 00:38:49,700 --> 00:38:52,990 So s of t will be integral from 0 801 00:38:52,990 --> 00:38:56,390 to t square root of a prime a squared times 802 00:38:56,390 --> 00:38:59,825 b squared b tau-- d tau, yes. 803 00:38:59,825 --> 00:39:01,205 S of t will be, what? 804 00:39:01,205 --> 00:39:02,740 Who's helping me on that? 805 00:39:02,740 --> 00:39:05,040 Because I want you to be awake. 806 00:39:05,040 --> 00:39:05,985 Are you guys awake? 807 00:39:05,985 --> 00:39:07,334 [CLASS MURMURS] 808 00:39:07,334 --> 00:39:08,750 PROFESSOR: The square root of that 809 00:39:08,750 --> 00:39:14,190 is a constant gets out times t. 810 00:39:14,190 --> 00:39:18,935 So what did I tell you when it comes to these functions? 811 00:39:18,935 --> 00:39:21,600 I have to turn my head badly like that. 812 00:39:21,600 --> 00:39:23,960 This was the alpha t or s of t. 813 00:39:23,960 --> 00:39:31,685 And this was t of s, which is the inverse function. 814 00:39:31,685 --> 00:39:33,310 I'm not going to write anything stupid. 815 00:39:33,310 --> 00:39:36,750 But this is practically the inverse function of s of t. 816 00:39:36,750 --> 00:39:39,190 I told you it was easiest t do. 817 00:39:39,190 --> 00:39:40,370 Put it here. 818 00:39:40,370 --> 00:39:43,480 T has to be replaced by, in terms of s, 819 00:39:43,480 --> 00:39:45,930 by a certain expression. 820 00:39:45,930 --> 00:39:47,924 So who is t? 821 00:39:47,924 --> 00:39:51,860 And you will do that in no time in the exam. 822 00:39:51,860 --> 00:39:55,910 T pulled out from there will be just 823 00:39:55,910 --> 00:40:00,720 s over square root a squared plus b squared 824 00:40:00,720 --> 00:40:04,560 s over square root a squared plus b squared 825 00:40:04,560 --> 00:40:08,668 and s over square root. 826 00:40:08,668 --> 00:40:10,600 OK? 827 00:40:10,600 --> 00:40:13,420 So can I keep the notation out of s? 828 00:40:13,420 --> 00:40:14,590 No. 829 00:40:14,590 --> 00:40:18,850 It's not mathematically correct to keep r of s. 830 00:40:18,850 --> 00:40:21,460 Why do the books sometimes by using 831 00:40:21,460 --> 00:40:23,040 multiplication keep r of s? 832 00:40:23,040 --> 00:40:27,390 Because the books are not always rigorous. 833 00:40:27,390 --> 00:40:28,925 But I'm trying to be rigorous. 834 00:40:28,925 --> 00:40:30,960 This is an honors class. 835 00:40:30,960 --> 00:40:34,940 So How do I rewrite the whole thing? 836 00:40:34,940 --> 00:40:39,961 r of t, who is a function of s, t as a function of s 837 00:40:39,961 --> 00:40:45,770 was again s over square root a squared plus b squared 838 00:40:45,770 --> 00:40:49,250 will be renamed rho of s. 839 00:40:49,250 --> 00:40:51,030 And what is that? 840 00:40:51,030 --> 00:40:54,810 That is a of cosine of parentheses 841 00:40:54,810 --> 00:41:00,570 s over square root a squared r b squared, comma, 842 00:41:00,570 --> 00:41:06,240 a sine of s over square root a squared plus b squared 843 00:41:06,240 --> 00:41:12,894 and b times s over square root a squared plus b squared. 844 00:41:12,894 --> 00:41:14,510 So what have I done? 845 00:41:14,510 --> 00:41:16,221 Did I get my 20%? 846 00:41:16,221 --> 00:41:16,720 Yes. 847 00:41:16,720 --> 00:41:17,230 Why? 848 00:41:17,230 --> 00:41:19,480 Because I reparameterized the curve. 849 00:41:19,480 --> 00:41:21,920 Did I get my other 20%? 850 00:41:21,920 --> 00:41:25,680 Yes, because I told people who s of t was. 851 00:41:25,680 --> 00:41:32,980 So 20% for this box and 20% for this expression. 852 00:41:32,980 --> 00:41:34,600 So what have I done? 853 00:41:34,600 --> 00:41:39,100 On the same physical curve, I have slowed down, thank God. 854 00:41:39,100 --> 00:41:41,860 You say, finally, she's slowing down, right? 855 00:41:41,860 --> 00:41:42,860 I've changed this speed. 856 00:41:42,860 --> 00:41:46,270 857 00:41:46,270 --> 00:41:51,130 On the contrary, if a would be 4 and be would be 3, 858 00:41:51,130 --> 00:41:56,170 I increase my speed multiple five times, right? 859 00:41:56,170 --> 00:41:59,500 So you can go back and forth between s and t. 860 00:41:59,500 --> 00:42:02,820 What does s do compared to t? 861 00:42:02,820 --> 00:42:04,601 It increases the speed five times. 862 00:42:04,601 --> 00:42:05,100 Yes sir. 863 00:42:05,100 --> 00:42:06,992 STUDENT: So when you reparameterize, 864 00:42:06,992 --> 00:42:08,825 it's just basically the integral from 0 to t 865 00:42:08,825 --> 00:42:11,850 of whatever [INAUDIBLE] of tau is. 866 00:42:11,850 --> 00:42:14,210 PROFESSOR: Exactly. 867 00:42:14,210 --> 00:42:18,680 So my suggestion to all of you-- it took me a year 868 00:42:18,680 --> 00:42:21,170 to understand how to reparameterize 869 00:42:21,170 --> 00:42:24,950 because I was not smart enough to get it as a freshman. 870 00:42:24,950 --> 00:42:26,380 I got an A in that class. 871 00:42:26,380 --> 00:42:28,390 I didn't understand anything. 872 00:42:28,390 --> 00:42:31,808 As a sophomore, I really-- because sometimes, you know, 873 00:42:31,808 --> 00:42:36,180 you can get an A without understanding things in there. 874 00:42:36,180 --> 00:42:38,607 As a sophomore, I said, OK, what the heck 875 00:42:38,607 --> 00:42:39,860 was that reparameterization? 876 00:42:39,860 --> 00:42:42,770 I have to understand that because it bothers me. 877 00:42:42,770 --> 00:42:43,340 I went back. 878 00:42:43,340 --> 00:42:45,220 I took the book. 879 00:42:45,220 --> 00:42:48,080 I learned about reparameterization. 880 00:42:48,080 --> 00:42:50,540 Our book, I think, does a very good job 881 00:42:50,540 --> 00:42:52,070 when it comes to reparameterizing. 882 00:42:52,070 --> 00:42:57,540 So if you open the 10.2 and 10.4, you have to skip-- well, 883 00:42:57,540 --> 00:42:59,860 am I telling you to skip 10.3? 884 00:42:59,860 --> 00:43:01,100 That's about ballistics. 885 00:43:01,100 --> 00:43:03,970 If you're interested in dancing and all sorts of, 886 00:43:03,970 --> 00:43:08,440 like, how the bullet will be projected 887 00:43:08,440 --> 00:43:11,346 in this motion or that motion, you can learn that. 888 00:43:11,346 --> 00:43:14,084 Those are plane curves that are interested in physics 889 00:43:14,084 --> 00:43:14,750 and mathematics. 890 00:43:14,750 --> 00:43:18,912 But 10.3 is not part of them and they are required. 891 00:43:18,912 --> 00:43:20,317 Read 10.2 and 10.4. 892 00:43:20,317 --> 00:43:21,650 You understand this much better. 893 00:43:21,650 --> 00:43:22,683 Yes, ma'am. 894 00:43:22,683 --> 00:43:24,891 STUDENT: Will the midterm or the final just be, like, 895 00:43:24,891 --> 00:43:26,682 a series problems, or will it be anything-- 896 00:43:26,682 --> 00:43:29,730 PROFESSOR: This is going to be like that-- 15 problems 897 00:43:29,730 --> 00:43:30,255 like that. 898 00:43:30,255 --> 00:43:31,963 STUDENT: Will it be anything, like, super 899 00:43:31,963 --> 00:43:33,214 in depth like the extra credit? 900 00:43:33,214 --> 00:43:35,090 PROFESSOR: That-- isn't that in-depth enough? 901 00:43:35,090 --> 00:43:37,390 OK, this is going to be like that. 902 00:43:37,390 --> 00:43:40,610 So I would say at this point, the way I feel, 903 00:43:40,610 --> 00:43:45,490 I feel that I am ready to put extra credit there. 904 00:43:45,490 --> 00:43:48,890 My policy is that I read everything. 905 00:43:48,890 --> 00:43:52,960 So even if at this point, you say extra credit. 906 00:43:52,960 --> 00:43:54,900 And you put it at the end for me. 907 00:43:54,900 --> 00:43:57,120 Say, look, I'm doing the extra credit here. 908 00:43:57,120 --> 00:44:00,350 Then I'll be ready and I'll say, OK, what did she mean? 909 00:44:00,350 --> 00:44:01,920 Length of the arc? 910 00:44:01,920 --> 00:44:02,420 Which arc? 911 00:44:02,420 --> 00:44:05,620 From here to here is ready to be computed. 912 00:44:05,620 --> 00:44:08,430 913 00:44:08,430 --> 00:44:11,290 And that's going to be-- you can include your extra credit 914 00:44:11,290 --> 00:44:13,340 inside the actual problem. 915 00:44:13,340 --> 00:44:14,259 I see it. 916 00:44:14,259 --> 00:44:14,800 STUDENT: Yes. 917 00:44:14,800 --> 00:44:15,670 PROFESSOR: Don't worry. 918 00:44:15,670 --> 00:44:17,336 STUDENT: Would it just be as like-- just 919 00:44:17,336 --> 00:44:19,720 like the casual problem on the test or midterm 920 00:44:19,720 --> 00:44:22,534 or whatever-- would it be, like, an extra credit 921 00:44:22,534 --> 00:44:23,478 problem in itself? 922 00:44:23,478 --> 00:44:24,894 I know there will be extra credit, 923 00:44:24,894 --> 00:44:26,310 but the kind of proving-- 924 00:44:26,310 --> 00:44:29,620 PROFESSOR: That is-- that is decided together 925 00:44:29,620 --> 00:44:32,830 with the course coordinator. 926 00:44:32,830 --> 00:44:35,100 The course coordinator himself said 927 00:44:35,100 --> 00:44:39,940 that he is encouraging us to set up the scale so that if you 928 00:44:39,940 --> 00:44:43,150 all the problems that are written on the exam, 929 00:44:43,150 --> 00:44:46,680 you get something like 120% if everything is perfect. 930 00:44:46,680 --> 00:44:47,860 STUDENT: OK, if we can-- 931 00:44:47,860 --> 00:44:49,738 PROFESSOR: So it's sort of in-built in that-- yes. 932 00:44:49,738 --> 00:44:51,220 STUDENT: If we can do the web work, 933 00:44:51,220 --> 00:44:53,200 is that a good indication of-- 934 00:44:53,200 --> 00:44:54,090 PROFESSOR: Wonderful. 935 00:44:54,090 --> 00:44:56,160 That's exactly-- because the way we 936 00:44:56,160 --> 00:44:58,290 write those problems for the final, 937 00:44:58,290 --> 00:45:01,830 we pull them out of the web work problems we do for homework. 938 00:45:01,830 --> 00:45:03,670 So a square root of a squared times 939 00:45:03,670 --> 00:45:08,195 b squared times pi over 2-- so what have I discovered? 940 00:45:08,195 --> 00:45:11,480 If I would take a piece of that paper 941 00:45:11,480 --> 00:45:13,760 and I would measure from this point to this point 942 00:45:13,760 --> 00:45:17,600 how much I traveled in inches from here to here, 943 00:45:17,600 --> 00:45:21,380 that's exactly that square root of- this would be like a 5. 944 00:45:21,380 --> 00:45:24,580 That's 3.1415 divided by 2. 945 00:45:24,580 --> 00:45:25,290 Yes, sir. 946 00:45:25,290 --> 00:45:29,340 STUDENT: So in the interval of a squared plus 947 00:45:29,340 --> 00:45:31,292 b squared, I know that that's supposed 948 00:45:31,292 --> 00:45:33,770 to be the interval the magnitude of r-- 949 00:45:33,770 --> 00:45:35,855 PROFESSOR: The speed-- integral of speed? 950 00:45:35,855 --> 00:45:36,480 STUDENT: Right. 951 00:45:36,480 --> 00:45:39,397 So which is the r prime, right? 952 00:45:39,397 --> 00:45:40,230 PROFESSOR: Yes, sir. 953 00:45:40,230 --> 00:45:43,667 STUDENT: OK, so r prime was-- 954 00:45:43,667 --> 00:45:44,500 PROFESSOR: Velocity. 955 00:45:44,500 --> 00:45:46,780 STUDENT: --a sine-- or negative a sine t, 956 00:45:46,780 --> 00:45:49,380 a cosine t, and then b? 957 00:45:49,380 --> 00:45:51,855 So where did the square root of a squared plus b squared 958 00:45:51,855 --> 00:45:53,487 come from? 959 00:45:53,487 --> 00:45:54,570 STUDENT: That's from the-- 960 00:45:54,570 --> 00:45:57,290 PROFESSOR: I just erased it. 961 00:45:57,290 --> 00:46:02,630 OK, so you have minus i-- minus a sine b equals sine p and d. 962 00:46:02,630 --> 00:46:04,970 When you squared them, what did you get? 963 00:46:04,970 --> 00:46:05,938 He has the same thing. 964 00:46:05,938 --> 00:46:06,979 STUDENT: So that's just-- 965 00:46:06,979 --> 00:46:08,831 PROFESSOR: The square of that plus the square root of that 966 00:46:08,831 --> 00:46:10,039 plus the square root of that. 967 00:46:10,039 --> 00:46:14,717 STUDENT: So it's just like a 2D representation of the top one. 968 00:46:14,717 --> 00:46:15,550 STUDENT: This side-- 969 00:46:15,550 --> 00:46:18,980 970 00:46:18,980 --> 00:46:21,365 PROFESSOR: I just need the magnitude of r prime, which 971 00:46:21,365 --> 00:46:22,880 is this p, right? 972 00:46:22,880 --> 00:46:23,657 STUDENT: Right. 973 00:46:23,657 --> 00:46:25,115 PROFESSOR: The magnitude of this is 974 00:46:25,115 --> 00:46:29,214 the speed, which is square root of a squared plus b squared. 975 00:46:29,214 --> 00:46:30,620 Is that clear? 976 00:46:30,620 --> 00:46:31,230 STUDENT: Yes. 977 00:46:31,230 --> 00:46:32,870 PROFESSOR: I can go on if you want. 978 00:46:32,870 --> 00:46:36,835 So a square root of-- the sum of the squares of this, this, 979 00:46:36,835 --> 00:46:40,510 and that is exactly square of [INAUDIBLE]. 980 00:46:40,510 --> 00:46:41,850 Keep this in mind as an example. 981 00:46:41,850 --> 00:46:44,710 It's an extremely important one. 982 00:46:44,710 --> 00:46:48,710 It appears very frequently in tests-- on tests. 983 00:46:48,710 --> 00:46:52,560 And it's one of the most beautiful examples 984 00:46:52,560 --> 00:46:57,900 in applications of mathematics to physics. 985 00:46:57,900 --> 00:47:02,740 I have something else that was there. 986 00:47:02,740 --> 00:47:03,570 Yes ma'am 987 00:47:03,570 --> 00:47:07,866 STUDENT: I was just going to ask if you want to curvature. 988 00:47:07,866 --> 00:47:08,480 PROFESSOR: Eh? 989 00:47:08,480 --> 00:47:09,265 STUDENT: The letter-- 990 00:47:09,265 --> 00:47:10,146 PROFESSOR: Curvature? 991 00:47:10,146 --> 00:47:11,020 STUDENT: Curvature. 992 00:47:11,020 --> 00:47:12,603 PROFESSOR: That's exactly what I want. 993 00:47:12,603 --> 00:47:17,970 And when I said I had something else for 20%, what was k? 994 00:47:17,970 --> 00:47:22,672 K was rho double prime of s in magnitude. 995 00:47:22,672 --> 00:47:29,784 So I have to be smart enough to look at that and rho of s. 996 00:47:29,784 --> 00:47:32,742 And rho of s was the thing that had 997 00:47:32,742 --> 00:47:35,693 here-- that's going to be probably the end of my lesson 998 00:47:35,693 --> 00:47:36,193 today. 999 00:47:36,193 --> 00:47:39,650 1000 00:47:39,650 --> 00:47:46,420 Since you have so many questions, I will continue. 1001 00:47:46,420 --> 00:47:50,360 I should consider-- the chapter is finished 1002 00:47:50,360 --> 00:47:54,030 but I will continue with a deeper review, how about that, 1003 00:47:54,030 --> 00:47:57,300 on Tuesday with more problems. 1004 00:47:57,300 --> 00:48:00,890 Because I have the feeling that although we covered 10.1, 10.2, 1005 00:48:00,890 --> 00:48:03,640 10.4, you need a lot more examples 1006 00:48:03,640 --> 00:48:05,790 until you feel comfortable. 1007 00:48:05,790 --> 00:48:08,350 Many of you not, maybe 10 people. 1008 00:48:08,350 --> 00:48:09,880 They feel very comfortable. 1009 00:48:09,880 --> 00:48:10,450 They get it. 1010 00:48:10,450 --> 00:48:13,330 But I think nobody will be hurt by more review and more 1011 00:48:13,330 --> 00:48:16,096 examples and more applications. 1012 00:48:16,096 --> 00:48:20,565 Now, who can help me finish my goal for today? 1013 00:48:20,565 --> 00:48:22,640 Is this hard? 1014 00:48:22,640 --> 00:48:25,580 This is rho of s. 1015 00:48:25,580 --> 00:48:30,360 So you have to tell me with the derivation, is it hard? 1016 00:48:30,360 --> 00:48:31,330 No. 1017 00:48:31,330 --> 00:48:37,530 Minus a sine of the whole thing times 1 1018 00:48:37,530 --> 00:48:40,551 over square root of a squared plus b squared because I'm 1019 00:48:40,551 --> 00:48:42,280 applying the chain rule, right? 1020 00:48:42,280 --> 00:48:43,900 Let me change color. 1021 00:48:43,900 --> 00:48:45,070 Who's the next guy? 1022 00:48:45,070 --> 00:48:50,135 A Cosine of s over square root a squared plus b squared. 1023 00:48:50,135 --> 00:48:53,040 I'm now going to leave you this as an exercise 1024 00:48:53,040 --> 00:48:56,250 because you're going to haunt me back ask me why I got this. 1025 00:48:56,250 --> 00:48:58,810 So I want to make it very clear. 1026 00:48:58,810 --> 00:49:03,940 B times 1 over square root a squared by b squared. 1027 00:49:03,940 --> 00:49:06,380 So are we happy with this? 1028 00:49:06,380 --> 00:49:07,460 Is this understood? 1029 00:49:07,460 --> 00:49:10,980 It's a simple derivation of the philosophy. 1030 00:49:10,980 --> 00:49:12,530 We are not done. 1031 00:49:12,530 --> 00:49:15,300 We have to do the acceleration. 1032 00:49:15,300 --> 00:49:18,400 So the acceleration with respect to s 1033 00:49:18,400 --> 00:49:21,610 of this curve where s was the arc length parameter 1034 00:49:21,610 --> 00:49:24,300 is real easy to compute in the same way. 1035 00:49:24,300 --> 00:49:26,140 What is different? 1036 00:49:26,140 --> 00:49:29,880 I'm not going to write more explicitly 1037 00:49:29,880 --> 00:49:32,360 because this should be visible for everybody. 1038 00:49:32,360 --> 00:49:35,450 STUDENT: x [INAUDIBLE]. 1039 00:49:35,450 --> 00:49:38,850 PROFESSOR: Good, minus a over-- I'll 1040 00:49:38,850 --> 00:49:41,640 wait for you to simplify because I don't 1041 00:49:41,640 --> 00:49:43,280 want to pull two roots out. 1042 00:49:43,280 --> 00:49:44,160 STUDENT: A squared-- 1043 00:49:44,160 --> 00:49:45,659 PROFESSOR: A squared plus b squared. 1044 00:49:45,659 --> 00:49:47,215 And why is that, [INAUDIBLE]? 1045 00:49:47,215 --> 00:49:52,180 Because you have once and twice from the chain rule. 1046 00:49:52,180 --> 00:49:55,590 So again, I hope you guys don't have a problem with the chain 1047 00:49:55,590 --> 00:50:00,750 rule so I don't have to send you back to Calculus 1. 1048 00:50:00,750 --> 00:50:05,630 A over a squared times b squared with a minus-- why 1049 00:50:05,630 --> 00:50:06,280 with a minus? 1050 00:50:06,280 --> 00:50:07,040 Somebody explain. 1051 00:50:07,040 --> 00:50:09,190 STUDENT: Use the derivative of cosine. 1052 00:50:09,190 --> 00:50:13,360 PROFESSOR: There's a cosine and there's a minus sine. 1053 00:50:13,360 --> 00:50:15,844 From deriving, I have a minus and a sine. 1054 00:50:15,844 --> 00:50:21,510 1055 00:50:21,510 --> 00:50:25,470 And finally, thank God, the 0-- why 0? 1056 00:50:25,470 --> 00:50:31,310 Because I have a constant that I'm deriving with respect to s. 1057 00:50:31,310 --> 00:50:33,050 Is it hard to see what's up? 1058 00:50:33,050 --> 00:50:36,110 What's going out? 1059 00:50:36,110 --> 00:50:40,814 What is the curvature of the helix? 1060 00:50:40,814 --> 00:50:45,230 A beautiful, beautiful function that 1061 00:50:45,230 --> 00:50:48,560 is known in most of these math, calculus, 1062 00:50:48,560 --> 00:50:54,070 multivariable calculus and differential geometry classes. 1063 00:50:54,070 --> 00:50:56,640 What did you get? 1064 00:50:56,640 --> 00:51:02,755 Square root of sum of the squares of all these guys. 1065 00:51:02,755 --> 00:51:04,115 You process it. 1066 00:51:04,115 --> 00:51:06,440 That's very easy. 1067 00:51:06,440 --> 00:51:07,395 Shall I write it down? 1068 00:51:07,395 --> 00:51:09,790 Let me write it down like a silly girl-- 1069 00:51:09,790 --> 00:51:13,500 square root of a squared, although I hate when I cannot 1070 00:51:13,500 --> 00:51:15,190 go ahead and simplify it. 1071 00:51:15,190 --> 00:51:18,950 But let's say there's this little baby thing. 1072 00:51:18,950 --> 00:51:21,550 1073 00:51:21,550 --> 00:51:25,070 Now I can say it's a over a squared 1074 00:51:25,070 --> 00:51:26,750 plus b squared-- finally. 1075 00:51:26,750 --> 00:51:29,106 So I'm going to ask you a few questions 1076 00:51:29,106 --> 00:51:30,480 and then I'm going to let you go. 1077 00:51:30,480 --> 00:51:32,840 It's a punishment for one minute. 1078 00:51:32,840 --> 00:51:36,580 OK, if I have the curve we had before, 1079 00:51:36,580 --> 00:51:41,590 the beautiful helix with a Pythagorean number 1080 00:51:41,590 --> 00:51:45,440 like 3 cosine t, 3 sine t, and 4t, what 1081 00:51:45,440 --> 00:51:48,498 is the curvature of that helix? 1082 00:51:48,498 --> 00:51:49,800 STUDENT: 3 over 5-- 1083 00:51:49,800 --> 00:51:52,246 PROFESSOR: 3 over 5, excellent. 1084 00:51:52,246 --> 00:51:53,730 How about my helix? 1085 00:51:53,730 --> 00:51:57,540 What if I changed the numbers in web work or on the midterm 1086 00:51:57,540 --> 00:52:00,600 and I say it's going to be-- it could even 1087 00:52:00,600 --> 00:52:02,150 be with a minus, guys. 1088 00:52:02,150 --> 00:52:05,130 It's just the way you travel it would be different. 1089 00:52:05,130 --> 00:52:08,030 So whether I put plus minus here, 1090 00:52:08,030 --> 00:52:09,950 you will try on different examples. 1091 00:52:09,950 --> 00:52:13,150 Sometimes if we put minus here or minus here, 1092 00:52:13,150 --> 00:52:15,380 it really doesn't matter. 1093 00:52:15,380 --> 00:52:17,880 Let's say we have cosine t sine t and t. 1094 00:52:17,880 --> 00:52:22,044 What's the curvature of that parametrized curve? 1095 00:52:22,044 --> 00:52:23,010 1 over-- 1096 00:52:23,010 --> 00:52:23,976 STUDENT: 2. 1097 00:52:23,976 --> 00:52:26,620 PROFESSOR: 1 over 2-- excellent. 1098 00:52:26,620 --> 00:52:27,380 So you got it. 1099 00:52:27,380 --> 00:52:28,740 So I'm proud of you. 1100 00:52:28,740 --> 00:52:32,396 Now, I want to do more examples until you feel confident 1101 00:52:32,396 --> 00:52:32,895 about it. 1102 00:52:32,895 --> 00:52:36,590 I know I got most of you to the point where I want it. 1103 00:52:36,590 --> 00:52:38,690 But you need more reading definitely 1104 00:52:38,690 --> 00:52:41,360 and you need to see more examples. 1105 00:52:41,360 --> 00:52:43,020 Feel free to read the whole chapter. 1106 00:52:43,020 --> 00:52:48,340 I would-- if you don't have time for 10.3, skip it. 1107 00:52:48,340 --> 00:52:50,140 10.5 is not going to be required. 1108 00:52:50,140 --> 00:52:53,090 So if I were a student, I'd go home, open the book, 1109 00:52:53,090 --> 00:52:55,860 read 10.1, 10.2, 10.4, close the book. 1110 00:52:55,860 --> 00:52:59,636 It's actually a lot less than you think it is. 1111 00:52:59,636 --> 00:53:01,510 If you go over the most important formulas, 1112 00:53:01,510 --> 00:53:04,230 then you are ready for the homework. 1113 00:53:04,230 --> 00:53:06,470 The second homework is due when? 1114 00:53:06,470 --> 00:53:08,090 February 11. 1115 00:53:08,090 --> 00:53:10,060 You guys have plenty of time. 1116 00:53:10,060 --> 00:53:14,260 Rather than going to the tutors, ask me for Tuesday. 1117 00:53:14,260 --> 00:53:17,420 On Tuesday, you'll have plenty of time for applications. 1118 00:53:17,420 --> 00:53:20,030 OK, have a wonderful weekend. 1119 00:53:20,030 --> 00:53:23,320 Don't forget to email when you get in trouble, OK? 1120 00:53:23,320 --> 00:53:27,696