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 --> 00:00:06,310 Your support will help MIT OpenCourseWare 4 00:00:06,310 --> 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 --> 00:00:17,116 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,116 --> 00:00:17,740 at ocw.mit.edu. 8 00:00:53,386 --> 00:00:54,300 PROFESSOR: Hi. 9 00:00:54,300 --> 00:00:58,050 Today we embark on a new phase of our course, 10 00:00:58,050 --> 00:01:01,130 signified by the beginning of block two. 11 00:01:01,130 --> 00:01:03,870 Where in block one, after indicating what 12 00:01:03,870 --> 00:01:07,570 mathematical structure was, in our quest 13 00:01:07,570 --> 00:01:11,000 to get as meaningfully as possible into the question 14 00:01:11,000 --> 00:01:12,930 of functions of several variables, 15 00:01:12,930 --> 00:01:15,690 we introduced the study of vectors. 16 00:01:15,690 --> 00:01:17,500 Vectors were important in their own right, 17 00:01:17,500 --> 00:01:20,970 so we paid some attention to that particular topic. 18 00:01:20,970 --> 00:01:24,420 And now we're at the next plateau, where one introduces 19 00:01:24,420 --> 00:01:27,580 the concept of functions. 20 00:01:27,580 --> 00:01:30,270 In other words, just like in ordinary arithmetic, 21 00:01:30,270 --> 00:01:33,260 after we get through with the arithmetic of constants, which 22 00:01:33,260 --> 00:01:35,600 is what you could basically call elementary school 23 00:01:35,600 --> 00:01:37,840 arithmetic-- you work with fixed numbers-- 24 00:01:37,840 --> 00:01:39,550 we then move into algebra. 25 00:01:39,550 --> 00:01:41,670 And from algebra we move into calculus. 26 00:01:41,670 --> 00:01:45,680 Sooner or later, we would like to study the calculus of vector 27 00:01:45,680 --> 00:01:47,130 functions and the like. 28 00:01:47,130 --> 00:01:49,850 And why not sooner? 29 00:01:49,850 --> 00:01:52,550 Which is what brings us to today's lesson. 30 00:01:52,550 --> 00:01:55,850 We are going to talk about a rather difficult mouthful, 31 00:01:55,850 --> 00:01:56,350 I guess. 32 00:01:56,350 --> 00:02:01,670 I call today's lecture "Vector Functions of Scalar Variables." 33 00:02:01,670 --> 00:02:04,470 And if that sounds like a tongue twister, 34 00:02:04,470 --> 00:02:07,950 let me point out that what has happened now is the following. 35 00:02:07,950 --> 00:02:10,479 When we did part one of this course, 36 00:02:10,479 --> 00:02:13,580 when we talked about functions and function machines, 37 00:02:13,580 --> 00:02:17,040 recall that both the input and the output of our machine 38 00:02:17,040 --> 00:02:20,352 were, by definition, scalars-- in other words, real numbers. 39 00:02:20,352 --> 00:02:21,810 Remember, we talked about functions 40 00:02:21,810 --> 00:02:23,990 of a single real variable. 41 00:02:23,990 --> 00:02:29,380 Now we have at our disposal both vectors and scalars. 42 00:02:29,380 --> 00:02:33,690 Consequently, the input can be either a vector or a scalar, 43 00:02:33,690 --> 00:02:37,580 and the output can be either a vector or a scalar. 44 00:02:37,580 --> 00:02:41,380 And this gives us an almost endless number 45 00:02:41,380 --> 00:02:44,690 of ways of combining these, where by "almost endless," 46 00:02:44,690 --> 00:02:45,890 I mean four. 47 00:02:45,890 --> 00:02:48,190 And the reason I say "almost endless" is this. 48 00:02:48,190 --> 00:02:49,340 There are only four ways. 49 00:02:49,340 --> 00:02:50,470 What are the four ways? 50 00:02:50,470 --> 00:02:52,100 Well, the four ways are what? 51 00:02:52,100 --> 00:02:54,120 Imagine that x is a scalar. 52 00:02:54,120 --> 00:02:56,080 In other words, the input is a scalar. 53 00:02:56,080 --> 00:02:58,400 Then the two possibilities are what? 54 00:02:58,400 --> 00:03:01,470 That the output is either a scalar or a vector. 55 00:03:01,470 --> 00:03:02,910 On the other hand, the input could 56 00:03:02,910 --> 00:03:06,810 have been a vector, in which case, there would have been 57 00:03:06,810 --> 00:03:08,510 two additional possibilities. 58 00:03:08,510 --> 00:03:11,500 Namely, the output would be either a scalar or a vector. 59 00:03:11,500 --> 00:03:14,890 Now the reason I call these four possibilities endless, 60 00:03:14,890 --> 00:03:17,630 or nearly endless, is the fact that by the time 61 00:03:17,630 --> 00:03:21,180 we get through discussing all four possible cases, 62 00:03:21,180 --> 00:03:23,440 the course will be essentially over. 63 00:03:23,440 --> 00:03:26,720 And to give you some hindsight as to what I mean, 64 00:03:26,720 --> 00:03:29,580 remember that we had a rather long course that 65 00:03:29,580 --> 00:03:31,470 was part one of this course. 66 00:03:31,470 --> 00:03:33,760 And notice that part one of this course 67 00:03:33,760 --> 00:03:38,040 was concerned only with that one case in four 68 00:03:38,040 --> 00:03:40,630 where both the input and the output 69 00:03:40,630 --> 00:03:43,480 happen to be real numbers, scalars. 70 00:03:43,480 --> 00:03:45,660 And it took us that long to develop the subject 71 00:03:45,660 --> 00:03:46,410 called what? 72 00:03:46,410 --> 00:03:49,900 Real functions of a single real variable-- input 73 00:03:49,900 --> 00:03:52,150 a scalar, output a scalar. 74 00:03:52,150 --> 00:03:57,610 Now the one that we've chosen for our next block, 75 00:03:57,610 --> 00:03:59,250 next combination that we've chosen, 76 00:03:59,250 --> 00:04:02,790 is where the input will be a scalar and the output 77 00:04:02,790 --> 00:04:05,980 will a vector. 78 00:04:05,980 --> 00:04:08,010 And in terms of physical examples, 79 00:04:08,010 --> 00:04:10,630 this can be made very, very meaningful. 80 00:04:10,630 --> 00:04:14,050 For example, one very common physical situation 81 00:04:14,050 --> 00:04:17,280 is that when we study force on an object-- force 82 00:04:17,280 --> 00:04:21,160 is obviously a vector-- but that the force on an object 83 00:04:21,160 --> 00:04:23,710 usually varies with time. 84 00:04:23,710 --> 00:04:26,330 But time happens to be a scalar. 85 00:04:26,330 --> 00:04:29,080 In other words, we might, for example, 86 00:04:29,080 --> 00:04:33,950 write that the vector F is a function of the scalar t. 87 00:04:33,950 --> 00:04:37,200 And rather than talk abstractly, let me make up 88 00:04:37,200 --> 00:04:39,350 a pseudo-physical situation. 89 00:04:39,350 --> 00:04:41,600 By pseudo-physical, I mean I haven't 90 00:04:41,600 --> 00:04:44,160 the vaguest notion where this formula would come up 91 00:04:44,160 --> 00:04:45,380 in real life. 92 00:04:45,380 --> 00:04:47,590 And I think that after you saw my performance 93 00:04:47,590 --> 00:04:51,000 on the explanation of work equals force times distance 94 00:04:51,000 --> 00:04:54,750 and why objects don't rise under the influence of friction 95 00:04:54,750 --> 00:04:56,160 and what have you, you believe me 96 00:04:56,160 --> 00:04:59,740 when I tell you I have no feel for these physical things. 97 00:04:59,740 --> 00:05:01,320 But I say that semi-jokingly. 98 00:05:01,320 --> 00:05:04,020 The main reason is that it's irrelevant what 99 00:05:04,020 --> 00:05:05,990 the physical application is. 100 00:05:05,990 --> 00:05:07,630 The important point is that each of you 101 00:05:07,630 --> 00:05:10,250 will find physical applications in your own way. 102 00:05:10,250 --> 00:05:13,240 All we really need is some generic problem 103 00:05:13,240 --> 00:05:16,630 that somehow signifies what all other problems look like. 104 00:05:16,630 --> 00:05:18,320 What I'm driving at here is, imagine 105 00:05:18,320 --> 00:05:20,790 that we have a force F, which varies 106 00:05:20,790 --> 00:05:24,710 with time, written in Cartesian coordinates as follows. 107 00:05:24,710 --> 00:05:29,710 The force is e to the minus t times i plus j. 108 00:05:29,710 --> 00:05:31,890 Notice that t is a scalar. 109 00:05:31,890 --> 00:05:36,380 As t varies, the scalar e to the minus t varies. 110 00:05:36,380 --> 00:05:39,110 But the scalar e to the minus t varying 111 00:05:39,110 --> 00:05:43,920 means that the vector e to the minus t i is changing, 112 00:05:43,920 --> 00:05:44,930 is varying. 113 00:05:44,930 --> 00:05:49,520 In other words, notice that e to the minus t i plus j 114 00:05:49,520 --> 00:05:53,870 is a variable vector, even though t is a scalar. 115 00:05:53,870 --> 00:05:57,300 For example, when t happens to be 0, what is the force? 116 00:05:57,300 --> 00:05:59,710 When t is 0, this would read what? 117 00:05:59,710 --> 00:06:03,550 The force is e to the minus 0 i plus j. 118 00:06:03,550 --> 00:06:05,890 That's the same as saying-- since e to the minus 0 119 00:06:05,890 --> 00:06:08,950 is 1-- the force, when the time is 0, 120 00:06:08,950 --> 00:06:11,890 is i plus j, which is a vector. 121 00:06:11,890 --> 00:06:14,540 You see, that vector changes as t changes, 122 00:06:14,540 --> 00:06:17,530 but t happens to be a scalar. 123 00:06:17,530 --> 00:06:21,030 Now here's the interesting point or an interesting point. 124 00:06:21,030 --> 00:06:23,310 I look at this expression, and I'm tempted 125 00:06:23,310 --> 00:06:25,490 to say something like this. 126 00:06:25,490 --> 00:06:31,310 I say for large values of t, F of t is approximately j. 127 00:06:31,310 --> 00:06:33,120 Now how did I arrive at that? 128 00:06:33,120 --> 00:06:34,890 Well what I said was, is I said, you know, 129 00:06:34,890 --> 00:06:39,040 e to the minus t gets very close to 0 as t gets large. 130 00:06:39,040 --> 00:06:44,500 As t gets large, therefore, this component tends to 0. 131 00:06:44,500 --> 00:06:48,970 This component stays constantly j-- or the component is 1, 132 00:06:48,970 --> 00:06:50,860 the vector is j. 133 00:06:50,860 --> 00:06:55,250 So that as t gets large, F of t behaves like j. 134 00:06:55,250 --> 00:06:58,930 And what I'm leading up to is that intuitively, 135 00:06:58,930 --> 00:07:01,040 I am now using the limit concept. 136 00:07:01,040 --> 00:07:02,560 I'm saying to myself, what? 137 00:07:02,560 --> 00:07:05,110 The limit of F of t as t approaches 138 00:07:05,110 --> 00:07:08,050 infinity et cetera is the limit of this sum. 139 00:07:08,050 --> 00:07:10,260 The limit of the sum is the sum of the limits. 140 00:07:10,260 --> 00:07:12,680 And I now take the limit term by term. 141 00:07:12,680 --> 00:07:16,870 And I sense that this limit is 0 and that this one is 1. 142 00:07:16,870 --> 00:07:18,920 Now you see, the whole point is this. 143 00:07:18,920 --> 00:07:23,830 What I would like to do is to study vector calculus. 144 00:07:23,830 --> 00:07:25,010 Meaning in this case, what? 145 00:07:25,010 --> 00:07:30,200 The calculus of vector functions of a scalar variable, 146 00:07:30,200 --> 00:07:32,180 of a scalar input. 147 00:07:32,180 --> 00:07:36,610 Now here's why structure was so important in our course. 148 00:07:36,610 --> 00:07:39,600 What I already know how to do is study limits 149 00:07:39,600 --> 00:07:43,320 in the case where both my input and my output were scalars. 150 00:07:43,320 --> 00:07:45,180 What I would like to be able to do 151 00:07:45,180 --> 00:07:49,790 is structurally inherit that entire system. 152 00:07:49,790 --> 00:07:51,150 Because it's a beautiful system. 153 00:07:51,150 --> 00:07:52,460 I understand it well. 154 00:07:52,460 --> 00:07:54,640 I have a whole bunch of theorems in that system. 155 00:07:54,640 --> 00:07:58,270 If I could only incorporate that verbatim into vector 156 00:07:58,270 --> 00:08:01,540 calculus, not only would I have a unifying thread, 157 00:08:01,540 --> 00:08:05,930 but I can save weeks of work not having to re-derive theorems 158 00:08:05,930 --> 00:08:09,339 for vectors which automatically have to be true because 159 00:08:09,339 --> 00:08:10,130 of their structure. 160 00:08:10,130 --> 00:08:12,980 Now that's, again, a difficult mouthful. 161 00:08:12,980 --> 00:08:15,770 This is written up voluminously in our notes. 162 00:08:15,770 --> 00:08:17,970 It's emphasized in the exercises, 163 00:08:17,970 --> 00:08:21,040 but I thought that I would try to show you a little bit live 164 00:08:21,040 --> 00:08:22,580 what this thing really means. 165 00:08:22,580 --> 00:08:24,330 Because I think that somehow or other, 166 00:08:24,330 --> 00:08:28,680 you have to hear these ideas, rather than just read them, 167 00:08:28,680 --> 00:08:30,640 to get the true feeling. 168 00:08:30,640 --> 00:08:33,919 What we do, for example, is we re-visit limits. 169 00:08:33,919 --> 00:08:36,200 And we write down the definition of limit 170 00:08:36,200 --> 00:08:39,409 just as it appeared in the scalar case. 171 00:08:39,409 --> 00:08:41,950 And let's start with an intuitive approach first, 172 00:08:41,950 --> 00:08:44,890 and later on, we'll get to the epsilon-delta approach. 173 00:08:44,890 --> 00:08:47,460 We said that the limit of f of x as x approaches 174 00:08:47,460 --> 00:08:53,090 a equals L means that f of x is near L 175 00:08:53,090 --> 00:08:55,690 provided that x is "near" a. 176 00:08:55,690 --> 00:08:58,930 And I've written the word "near" in quotation marks here 177 00:08:58,930 --> 00:09:02,632 to emphasize that we were using a geometrical phrase-- well, 178 00:09:02,632 --> 00:09:04,340 I guess you can't call one word a phrase, 179 00:09:04,340 --> 00:09:08,180 but a geometrical term-- to emphasize an arithmetic 180 00:09:08,180 --> 00:09:08,720 concept. 181 00:09:08,720 --> 00:09:13,530 Namely, to say that f of x was near L where f of x and L 182 00:09:13,530 --> 00:09:17,200 were numbers meant quite simply that what? 183 00:09:17,200 --> 00:09:20,710 The numerical value of f of x was very nearly equal 184 00:09:20,710 --> 00:09:24,110 to the numerical value of L. And that's the same as saying what? 185 00:09:24,110 --> 00:09:28,740 That the absolute value of the difference f of x minus L 186 00:09:28,740 --> 00:09:31,010 was very small. 187 00:09:31,010 --> 00:09:32,830 I just mention that, OK? 188 00:09:32,830 --> 00:09:37,110 Now what we would like to do is invent a definition of limit 189 00:09:37,110 --> 00:09:40,400 for a vector-valued function of a scalar variable. 190 00:09:40,400 --> 00:09:42,400 One way of doing this is of course 191 00:09:42,400 --> 00:09:45,240 to make up a completely brand new definition that has 192 00:09:45,240 --> 00:09:47,160 nothing to do with the past. 193 00:09:47,160 --> 00:09:49,380 But as in every field of human endeavor, 194 00:09:49,380 --> 00:09:54,480 one likes to plan the future and further sorties 195 00:09:54,480 --> 00:09:58,530 after one has modeled the successes of the past. 196 00:09:58,530 --> 00:10:01,210 So what we do is a very simple device 197 00:10:01,210 --> 00:10:02,980 and yet very, very powerful. 198 00:10:02,980 --> 00:10:07,340 We vectorize the definition that has already served us so well. 199 00:10:07,340 --> 00:10:09,740 What I mean by that is, I go back 200 00:10:09,740 --> 00:10:12,800 to the definition which I've written here and put in vectors 201 00:10:12,800 --> 00:10:14,260 in appropriate places. 202 00:10:14,260 --> 00:10:17,490 For example, we're dealing with a vector function, 203 00:10:17,490 --> 00:10:20,040 so f has to have an arrow over it. 204 00:10:20,040 --> 00:10:23,130 Moreover, since f of x is a vector, 205 00:10:23,130 --> 00:10:26,100 anything that it approaches as a limit by definition 206 00:10:26,100 --> 00:10:27,860 must also be a vector. 207 00:10:27,860 --> 00:10:31,490 So that means the L must also be arrowized. 208 00:10:31,490 --> 00:10:32,810 All right? 209 00:10:32,810 --> 00:10:34,710 Put an arrow over the L. 210 00:10:34,710 --> 00:10:36,660 On the other hand, the x and the a 211 00:10:36,660 --> 00:10:40,890 remain left alone because after all, they are just scalars. 212 00:10:40,890 --> 00:10:46,010 Remember again, x and a are inputs of our function machine. 213 00:10:46,010 --> 00:10:47,560 And in the particular investigation 214 00:10:47,560 --> 00:10:50,430 that we're making, our input happens to be a scalar. 215 00:10:50,430 --> 00:10:52,850 It's only the output that's a vector. 216 00:10:52,850 --> 00:10:55,630 So let's go back and, just as I say, arrowize 217 00:10:55,630 --> 00:10:59,230 or vectorize whatever has to be vectorized here. 218 00:10:59,230 --> 00:11:01,990 So I now have a new definition, intuitively. 219 00:11:01,990 --> 00:11:03,650 The limit of the vector function f 220 00:11:03,650 --> 00:11:08,680 of x as x approaches a is the vector L means that the vector 221 00:11:08,680 --> 00:11:12,725 f of x is near the vector L, provided that the scalar x 222 00:11:12,725 --> 00:11:14,720 is near the scalar a. 223 00:11:14,720 --> 00:11:18,220 Now the point is that the "x is near a" 224 00:11:18,220 --> 00:11:20,770 doesn't need any further interpretation because a 225 00:11:20,770 --> 00:11:22,510 and x are still scalars. 226 00:11:22,510 --> 00:11:24,770 The question that comes up now is twofold. 227 00:11:24,770 --> 00:11:29,070 First of all, does it make sense to say that f of x is near L? 228 00:11:29,070 --> 00:11:31,950 Does that even make sense when you're talking about vectors? 229 00:11:31,950 --> 00:11:35,120 The second thing is, if it does make sense, 230 00:11:35,120 --> 00:11:37,110 is it the meaning that we want it to have? 231 00:11:37,110 --> 00:11:39,970 Does it capture our intuitive feeling? 232 00:11:39,970 --> 00:11:43,550 Let's take the questions in order of appearance. 233 00:11:43,550 --> 00:11:47,300 First of all, what does it mean to say that f of x-- the vector 234 00:11:47,300 --> 00:11:49,690 f of x-- is near the vector L? 235 00:11:49,690 --> 00:11:51,970 In fact, let's generalize that. 236 00:11:51,970 --> 00:11:54,210 Given any two vectors A and B, what 237 00:11:54,210 --> 00:11:59,600 does it mean to say that the vector A is near the vector B? 238 00:11:59,600 --> 00:12:01,270 Now what my claim is, is this. 239 00:12:01,270 --> 00:12:04,504 Let's draw the vectors A and B as arrows in the plane here. 240 00:12:04,504 --> 00:12:06,920 Let's assume that they start at a common origin, which you 241 00:12:06,920 --> 00:12:08,850 can always assume, of course. 242 00:12:08,850 --> 00:12:13,640 To say that A is nearly equal to B somehow means what? 243 00:12:13,640 --> 00:12:17,520 That the vector A should be very nearly equal to the vector B. 244 00:12:17,520 --> 00:12:20,060 Well remember, if two vectors start 245 00:12:20,060 --> 00:12:22,120 at a common point, the only way that they 246 00:12:22,120 --> 00:12:25,410 can be equal is if their heads coincide. 247 00:12:25,410 --> 00:12:28,820 Consequently, with A and B starting at a common point, 248 00:12:28,820 --> 00:12:31,840 to say that A is nearly equal to B 249 00:12:31,840 --> 00:12:34,340 should mean that the distance between the head of A 250 00:12:34,340 --> 00:12:36,890 and the head of B should be small. 251 00:12:36,890 --> 00:12:39,530 But how do we state the distance between the head of A 252 00:12:39,530 --> 00:12:40,570 and the head of B? 253 00:12:40,570 --> 00:12:44,130 Notice that that has a very convenient numerical form. 254 00:12:44,130 --> 00:12:47,700 Namely, the vector that joins A to B 255 00:12:47,700 --> 00:12:50,400 can be written as either A minus B or B minus A, 256 00:12:50,400 --> 00:12:52,390 depending on what sense you put on this. 257 00:12:52,390 --> 00:12:53,720 We don't care about the sense. 258 00:12:53,720 --> 00:12:55,650 All we care about is the magnitude. 259 00:12:55,650 --> 00:13:00,000 Let's just call this length the magnitude of A minus B. 260 00:13:00,000 --> 00:13:01,230 And now we're in business. 261 00:13:01,230 --> 00:13:03,190 Namely, we say, lookit. 262 00:13:03,190 --> 00:13:07,800 To say that A is near B means that A is nearly equal to B, 263 00:13:07,800 --> 00:13:13,190 which in turn means that the magnitude of A minus B-- 264 00:13:13,190 --> 00:13:16,110 the magnitude of the vector-- what vector?-- 265 00:13:16,110 --> 00:13:19,290 the vector A minus the vector B-- is small. 266 00:13:19,290 --> 00:13:22,110 And keep in mind that I'm now using 'small' 267 00:13:22,110 --> 00:13:23,840 as I did arithmetically. 268 00:13:23,840 --> 00:13:26,530 Because even though A and B are vectors, 269 00:13:26,530 --> 00:13:31,000 and so also is A minus B, the magnitude of a vector 270 00:13:31,000 --> 00:13:32,320 is a number. 271 00:13:32,320 --> 00:13:34,940 Note this very important thing. 272 00:13:34,940 --> 00:13:36,580 Even when I'm dealing with vectors, 273 00:13:36,580 --> 00:13:40,580 as soon as I mention magnitude, I'm talking about a number. 274 00:13:40,580 --> 00:13:42,690 In other words then, I now define 275 00:13:42,690 --> 00:13:47,320 A to be near B to mean that the magnitude of the difference 276 00:13:47,320 --> 00:13:51,690 between A and B is small. 277 00:13:51,690 --> 00:13:54,400 And again, question two-- does that 278 00:13:54,400 --> 00:13:56,320 caption my intuitive feeling? 279 00:13:56,320 --> 00:13:57,710 And the answer is, yes, it does. 280 00:13:57,710 --> 00:14:03,500 I want A to be near B to mean that they nearly coincide. 281 00:14:03,500 --> 00:14:06,490 And that is the same as saying that the magnitude of A minus B 282 00:14:06,490 --> 00:14:07,380 is small. 283 00:14:07,380 --> 00:14:09,730 At any rate, having gone through this 284 00:14:09,730 --> 00:14:11,790 from a fairly intuitive point of view, 285 00:14:11,790 --> 00:14:15,720 let's now revisit limits more rigorously and rewrite 286 00:14:15,720 --> 00:14:19,414 our limit definition in terms of epsilons and deltas. 287 00:14:19,414 --> 00:14:21,830 And by the way, I hope this doesn't look like anything new 288 00:14:21,830 --> 00:14:24,060 to you, what I'm going to be talking about next. 289 00:14:24,060 --> 00:14:26,380 It's our old definition of limit that 290 00:14:26,380 --> 00:14:29,540 was the backbone of the entire part one of this course. 291 00:14:29,540 --> 00:14:31,960 I not only hope that it looks familiar to you, 292 00:14:31,960 --> 00:14:35,800 I hope that you read this thing almost subconsciously 293 00:14:35,800 --> 00:14:37,520 as second nature by now. 294 00:14:37,520 --> 00:14:38,620 But the statement is what? 295 00:14:38,620 --> 00:14:42,120 The limit of f of x as x approaches a equals L means, 296 00:14:42,120 --> 00:14:44,430 given any epsilon greater than 0, 297 00:14:44,430 --> 00:14:47,520 we can find a delta greater than 0 such 298 00:14:47,520 --> 00:14:52,040 that whenever the absolute value of x minus a is less than delta 299 00:14:52,040 --> 00:14:54,340 but greater than 0, the implication 300 00:14:54,340 --> 00:14:57,670 is that the absolute value of f of x minus L 301 00:14:57,670 --> 00:14:59,510 is less than epsilon. 302 00:14:59,510 --> 00:15:02,730 Now if I go back to my structural properties again, 303 00:15:02,730 --> 00:15:06,340 it seems to me that what my first approximation, at least-- 304 00:15:06,340 --> 00:15:10,150 for a rigorous definition to limits of vector functions-- 305 00:15:10,150 --> 00:15:12,330 should be to just, as I did before, 306 00:15:12,330 --> 00:15:14,930 read through this definition verbatim. 307 00:15:14,930 --> 00:15:17,890 Don't change a thing, but just vectorize 308 00:15:17,890 --> 00:15:19,970 what has to be vectorized. 309 00:15:19,970 --> 00:15:22,740 And because this definition that I'm going to give 310 00:15:22,740 --> 00:15:26,000 is so important, I elect to rewrite it. 311 00:15:26,000 --> 00:15:27,740 And what you're going to see next 312 00:15:27,740 --> 00:15:30,390 is nothing more than a carbon copy, 313 00:15:30,390 --> 00:15:34,570 so to speak, of this, only with appropriate arrows being 314 00:15:34,570 --> 00:15:35,490 put in. 315 00:15:35,490 --> 00:15:39,150 Namely, I am going to give, as my rigorous definition 316 00:15:39,150 --> 00:15:41,350 of the limit of f of x as x approaches 317 00:15:41,350 --> 00:15:44,920 a equals the vector L-- it's going to mean what? 318 00:15:44,920 --> 00:15:48,950 Given epsilon greater than 0, I can find delta greater 319 00:15:48,950 --> 00:15:53,240 than 0 such that whenever the absolute value of x minus a 320 00:15:53,240 --> 00:15:57,340 is less than delta but greater than 0, the implication is 321 00:15:57,340 --> 00:16:00,175 that the magnitude-- see I don't say absolute value now, 322 00:16:00,175 --> 00:16:02,050 because notice, I'm not dealing with numbers. 323 00:16:02,050 --> 00:16:06,390 These are vectors-- but the magnitude of f of x minus L 324 00:16:06,390 --> 00:16:09,366 must be less than epsilon. 325 00:16:09,366 --> 00:16:13,260 Now you see what I've done: I've obtained the second definition 326 00:16:13,260 --> 00:16:16,830 from the first simply by appropriate vectorization. 327 00:16:16,830 --> 00:16:20,870 What I have to make sure of is what? 328 00:16:20,870 --> 00:16:22,760 That this thing still makes sense. 329 00:16:22,760 --> 00:16:24,380 And notice that it does. 330 00:16:24,380 --> 00:16:26,950 Notice that what this says, in plain English is, what? 331 00:16:26,950 --> 00:16:34,200 That if x is sufficiently close to a-- what? 332 00:16:34,200 --> 00:16:36,920 The magnitude of the difference between these two vectors 333 00:16:36,920 --> 00:16:38,410 is as small as we wish. 334 00:16:38,410 --> 00:16:41,289 But we call that magnitude what? 335 00:16:41,289 --> 00:16:42,830 The distance between the two vectors. 336 00:16:42,830 --> 00:16:45,020 This says what? 337 00:16:45,020 --> 00:16:49,160 Correlating our formal language with our informal language, 338 00:16:49,160 --> 00:16:50,640 this simply says what? 339 00:16:50,640 --> 00:16:54,170 That we can make f of x as near to L 340 00:16:54,170 --> 00:16:58,640 as we wish just by picking x sufficiently close to a. 341 00:16:58,640 --> 00:17:03,240 So now what that tells us is that the new limit definition 342 00:17:03,240 --> 00:17:06,626 makes sense, just by mimicking the old definition. 343 00:17:06,626 --> 00:17:08,000 And what do we mean by mimicking? 344 00:17:08,000 --> 00:17:11,180 We mean vectorize the old definition, 345 00:17:11,180 --> 00:17:13,730 go through it word for word and put in vectors 346 00:17:13,730 --> 00:17:15,579 in appropriate places. 347 00:17:15,579 --> 00:17:17,670 Now let's see what this means structurally. 348 00:17:17,670 --> 00:17:21,200 That's the whole key to our particular lecture today: 349 00:17:21,200 --> 00:17:23,109 the structural value of this. 350 00:17:23,109 --> 00:17:26,380 For example, what were some of the building blocks 351 00:17:26,380 --> 00:17:29,860 that we based calculus of a single variable on? 352 00:17:29,860 --> 00:17:33,160 Remember, derivative was defined as a limit, 353 00:17:33,160 --> 00:17:35,890 and therefore all of our properties about derivatives 354 00:17:35,890 --> 00:17:37,660 followed from limit properties. 355 00:17:37,660 --> 00:17:42,830 Well, among the limit properties that we had for part one where, 356 00:17:42,830 --> 00:17:43,330 what? 357 00:17:43,330 --> 00:17:45,220 Input and output were scalars. 358 00:17:45,220 --> 00:17:49,810 The limit of the sum was equal to the sum of the limits. 359 00:17:49,810 --> 00:17:52,480 And the limit of a product is equal to the product 360 00:17:52,480 --> 00:17:53,610 the limits. 361 00:17:53,610 --> 00:17:55,690 And remember, in all of our proofs, 362 00:17:55,690 --> 00:17:58,900 we use these particular structural properties. 363 00:17:58,900 --> 00:17:59,400 You see? 364 00:17:59,400 --> 00:18:01,020 The next thing we would like to know-- 365 00:18:01,020 --> 00:18:03,300 see after all, these structural properties, 366 00:18:03,300 --> 00:18:04,800 even though they're called theorems, 367 00:18:04,800 --> 00:18:07,290 essentially, they become the rules of the game 368 00:18:07,290 --> 00:18:08,560 once we've proven them. 369 00:18:08,560 --> 00:18:10,430 In other words, once we've proved 370 00:18:10,430 --> 00:18:14,050 using epsilons and deltas that this happens to be true, 371 00:18:14,050 --> 00:18:18,030 notice that we never again use the fact that these are 372 00:18:18,030 --> 00:18:22,100 epsilons and deltas, that we have epsilons and deltas when 373 00:18:22,100 --> 00:18:24,870 we use this particular result. We just write it down. 374 00:18:24,870 --> 00:18:27,730 Because once we've proven it, we can always use it. 375 00:18:27,730 --> 00:18:29,720 In other words, what I'm really saying is, 376 00:18:29,720 --> 00:18:31,440 let me vectorize this. 377 00:18:35,392 --> 00:18:37,640 And I'll put a question mark now, because you see, 378 00:18:37,640 --> 00:18:39,450 I don't know if this is true for vectors. 379 00:18:39,450 --> 00:18:40,867 I don't know if the limit of a sum 380 00:18:40,867 --> 00:18:42,325 is the sum of the limits when we're 381 00:18:42,325 --> 00:18:44,360 dealing with vectors rather than with scalars. 382 00:18:44,360 --> 00:18:45,980 But notice what I've done. 383 00:18:45,980 --> 00:18:51,020 I have structurally plagiarized from my original definition. 384 00:18:51,020 --> 00:18:53,069 All I did was I vectorized it. 385 00:18:53,069 --> 00:18:54,610 You see, what I'm going to have to do 386 00:18:54,610 --> 00:18:57,050 is to check from my original definition 387 00:18:57,050 --> 00:18:59,562 whether these properties still hold true. 388 00:18:59,562 --> 00:19:01,270 I'm going to talk about that in a minute. 389 00:19:01,270 --> 00:19:03,380 Let me just continue on for a while. 390 00:19:03,380 --> 00:19:05,430 See, similarly over here, we talked 391 00:19:05,430 --> 00:19:07,350 about the limit of a product equaling 392 00:19:07,350 --> 00:19:08,780 the product of the limits. 393 00:19:08,780 --> 00:19:10,840 So again, we'd like to use results like that. 394 00:19:10,840 --> 00:19:12,590 And somebody says, well, let's vectorize. 395 00:19:14,684 --> 00:19:16,350 I want to show you what I mean by saying 396 00:19:16,350 --> 00:19:19,389 that after you vectorize, you have to be darn careful 397 00:19:19,389 --> 00:19:21,430 that you don't just go around saying, look at it, 398 00:19:21,430 --> 00:19:22,090 this is legal. 399 00:19:22,090 --> 00:19:24,080 I put arrows over everything. 400 00:19:24,080 --> 00:19:26,410 Sure it's legal, but you may not have 401 00:19:26,410 --> 00:19:28,240 anything that's worth keeping. 402 00:19:28,240 --> 00:19:29,820 Look at this expression. 403 00:19:29,820 --> 00:19:33,030 What are f and g in this case? 404 00:19:33,030 --> 00:19:36,090 The way I've written it, these happen to be vectors. 405 00:19:36,090 --> 00:19:37,280 See, they're both arrowized. 406 00:19:37,280 --> 00:19:37,780 Right? 407 00:19:37,780 --> 00:19:38,870 They're vectors. 408 00:19:38,870 --> 00:19:41,110 Have we talked about the ordinary product 409 00:19:41,110 --> 00:19:42,650 of two vector functions? 410 00:19:42,650 --> 00:19:43,860 And the answer is no. 411 00:19:43,860 --> 00:19:45,930 When we have two vector functions, 412 00:19:45,930 --> 00:19:48,710 we must either have a dot or a cross 413 00:19:48,710 --> 00:19:51,210 in here-- the dot product or the cross product. 414 00:19:51,210 --> 00:19:53,210 And if we're not going to put a dot in here, 415 00:19:53,210 --> 00:19:57,190 the best we can talk about is scalar multiplication. 416 00:19:57,190 --> 00:20:00,500 In other words, let me-- before you tend to memorize this, 417 00:20:00,500 --> 00:20:02,370 let me cross that out. 418 00:20:02,370 --> 00:20:04,350 Because this is ambiguous. 419 00:20:04,350 --> 00:20:05,610 It doesn't make sense. 420 00:20:05,610 --> 00:20:07,610 And what I am saying is that unfortunately-- not 421 00:20:07,610 --> 00:20:09,943 unfortunately, but if I want to be rigorous about this-- 422 00:20:09,943 --> 00:20:12,430 there happens to be three interpretations here. 423 00:20:12,430 --> 00:20:14,190 Namely, I can do what? 424 00:20:14,190 --> 00:20:16,790 I can think of f(x) as being a scalar 425 00:20:16,790 --> 00:20:19,180 and g(x) as being a vector. 426 00:20:19,180 --> 00:20:24,190 Or I can think of f(x) and g(x) as both being vector functions, 427 00:20:24,190 --> 00:20:26,360 but in one case having the dot product 428 00:20:26,360 --> 00:20:28,180 and in the other case, the cross product. 429 00:20:28,180 --> 00:20:32,120 In other words, to vectorize this particular equation, 430 00:20:32,120 --> 00:20:37,800 I guess what I have to do is write down 431 00:20:37,800 --> 00:20:39,630 these three possibilities. 432 00:20:39,630 --> 00:20:41,950 Namely, notice in this case, this is what? 433 00:20:41,950 --> 00:20:45,360 This is a scalar multiple of a vector function. 434 00:20:45,360 --> 00:20:48,270 Granted that the scalar multiple is a variable in this case, 435 00:20:48,270 --> 00:20:49,300 this still makes sense. 436 00:20:49,300 --> 00:20:54,170 Namely, for a fixed x-- for a fixed x, f of x is a constant, 437 00:20:54,170 --> 00:20:55,830 and g of x is a vector. 438 00:20:55,830 --> 00:20:59,630 A constant times a vector is a vector. 439 00:20:59,630 --> 00:21:04,390 For a fixed x, f and g here are fixed vectors. 440 00:21:04,390 --> 00:21:07,320 And I can talk about the dot product of two vectors. 441 00:21:07,320 --> 00:21:11,050 And for a fixed x over here, f of x and g of x 442 00:21:11,050 --> 00:21:13,850 are again fixed vectors, and consequently, I 443 00:21:13,850 --> 00:21:15,630 can talk about their cross product. 444 00:21:15,630 --> 00:21:17,900 And the question is, will these properties still 445 00:21:17,900 --> 00:21:20,930 be true when we're dealing with vector functions? 446 00:21:20,930 --> 00:21:22,920 The answer happens to be yes. 447 00:21:22,920 --> 00:21:25,260 But it's not yes automatically. 448 00:21:25,260 --> 00:21:28,970 In other words, what we're going to do in the notes is 449 00:21:28,970 --> 00:21:31,770 to mimic-- to prove these results for vectors, what we 450 00:21:31,770 --> 00:21:35,610 are going to do is to mimic every single proof 451 00:21:35,610 --> 00:21:38,020 that we gave in the scalar case, only 452 00:21:38,020 --> 00:21:41,450 replacing scalars by appropriate vectors 453 00:21:41,450 --> 00:21:43,720 whenever this is supposed to happen. 454 00:21:43,720 --> 00:21:46,180 The trouble is that certain results which 455 00:21:46,180 --> 00:21:48,880 were true for scalars may not automatically 456 00:21:48,880 --> 00:21:50,710 be true for vectors. 457 00:21:50,710 --> 00:21:52,520 Let me give you an example. 458 00:21:52,520 --> 00:21:54,780 Somehow or other to prove that the limit of a sum 459 00:21:54,780 --> 00:21:56,400 was equal to the sum of the limits, 460 00:21:56,400 --> 00:21:59,470 we used the property of absolute values 461 00:21:59,470 --> 00:22:02,600 that said that the absolute value of a sum was less than 462 00:22:02,600 --> 00:22:05,290 or equal to the sum of the absolute values. 463 00:22:05,290 --> 00:22:07,130 And we proved that for numbers. 464 00:22:07,130 --> 00:22:09,150 Suppose I now vectorize this. 465 00:22:14,750 --> 00:22:15,480 See? 466 00:22:15,480 --> 00:22:17,530 This now means magnitudes, right? 467 00:22:17,530 --> 00:22:20,310 And if a and b are any vectors in the plane, 468 00:22:20,310 --> 00:22:23,090 a and b no longer have to be in the same direction-- 469 00:22:23,090 --> 00:22:27,390 how do we know that vector addition has the same magnitude 470 00:22:27,390 --> 00:22:30,240 property that scalar addition has? 471 00:22:30,240 --> 00:22:33,010 See, can we be sure that this recipe is true? 472 00:22:33,010 --> 00:22:36,510 Well let's go and see what this recipe means. 473 00:22:36,510 --> 00:22:39,400 By the way, this does happen to be true in vector arithmetic. 474 00:22:39,400 --> 00:22:42,130 And it happens to be known as the triangle inequality. 475 00:22:42,130 --> 00:22:44,870 And the reason for that is, if I draw a triangle calling 476 00:22:44,870 --> 00:22:47,700 two of the sides a and b, respectively, 477 00:22:47,700 --> 00:22:51,830 and the third side c, we do know from elementary geometry 478 00:22:51,830 --> 00:22:56,030 that the sum of the lengths of two sides of a triangle 479 00:22:56,030 --> 00:22:58,900 is greater than or equal to the length of the third side 480 00:22:58,900 --> 00:23:00,040 of the triangle. 481 00:23:00,040 --> 00:23:03,400 What I'm driving at now is-- let me just vectorize this and show 482 00:23:03,400 --> 00:23:05,480 you something that I think is very cute here. 483 00:23:05,480 --> 00:23:08,910 If I put arrows here, this becomes the vector a, now; 484 00:23:08,910 --> 00:23:12,896 this becomes the vector b; this becomes the vector c. 485 00:23:12,896 --> 00:23:13,750 OK? 486 00:23:13,750 --> 00:23:16,590 But in terms of our definition of vector addition, 487 00:23:16,590 --> 00:23:20,040 noticed that c goes from the tail of a to the head of b. 488 00:23:20,040 --> 00:23:23,380 a and b are lined up properly for addition. 489 00:23:23,380 --> 00:23:27,400 So this is the vector a plus b. 490 00:23:27,400 --> 00:23:29,940 Now state the triangle inequality 491 00:23:29,940 --> 00:23:31,620 in terms of this picture. 492 00:23:31,620 --> 00:23:34,280 The triangle inequality says what? 493 00:23:34,280 --> 00:23:37,330 The third side of the triangle-- well what length is this? 494 00:23:37,330 --> 00:23:39,630 If the vector is a plus b, the length 495 00:23:39,630 --> 00:23:43,540 is the magnitude of a plus b-- must be less than 496 00:23:43,540 --> 00:23:46,720 or equal to the sum of the lengths of the other two sides. 497 00:23:46,720 --> 00:23:52,730 But those are just this. 498 00:23:52,730 --> 00:23:55,960 And that verifies the recipe that we want. 499 00:23:55,960 --> 00:23:57,910 In other words, it's going to turn out 500 00:23:57,910 --> 00:24:03,060 that every vector property that we need-- what 501 00:24:03,060 --> 00:24:05,690 do we mean by need-- that happened to be true for scalar 502 00:24:05,690 --> 00:24:09,220 calculus-- is going to be sufficient in vector calculus 503 00:24:09,220 --> 00:24:09,910 as well. 504 00:24:09,910 --> 00:24:13,570 For example, when we proved the product rule for scalars-- 505 00:24:13,570 --> 00:24:15,570 not the product rule, but the limit of a product 506 00:24:15,570 --> 00:24:17,780 was the product of the limits-- we used such things 507 00:24:17,780 --> 00:24:20,770 as the absolute value of a product 508 00:24:20,770 --> 00:24:23,420 was equal to the product of the absolute values. 509 00:24:23,420 --> 00:24:26,330 Notice, for example, in terms of case one over here, 510 00:24:26,330 --> 00:24:31,080 if I vectorize b and leave a as a scalar, the magnitude 511 00:24:31,080 --> 00:24:35,330 of a times the vector b is equal to the magnitude 512 00:24:35,330 --> 00:24:38,730 of a times the magnitude of b, just by the definition of what 513 00:24:38,730 --> 00:24:40,350 we meant by a scalar multiple. 514 00:24:40,350 --> 00:24:42,600 After all, a times b meant what? 515 00:24:42,600 --> 00:24:45,670 You just kept the direction of b constant, 516 00:24:45,670 --> 00:24:50,840 but multiplied b by the magnitude of a. 517 00:24:50,840 --> 00:24:51,540 Meaning what? 518 00:24:51,540 --> 00:24:54,250 By a, if was a positive; minus a, if a was negative, 519 00:24:54,250 --> 00:24:55,975 this is certainly true. 520 00:24:55,975 --> 00:24:57,850 By the way, there is a very important caution 521 00:24:57,850 --> 00:24:59,470 that I'd like to warn you about. 522 00:24:59,470 --> 00:25:01,760 And this is done in great detail in the notes. 523 00:25:01,760 --> 00:25:06,650 It's very important to point out that when I vectorize this 524 00:25:06,650 --> 00:25:08,820 in terms of a dot and a cross product, 525 00:25:08,820 --> 00:25:12,510 it is not true that the magnitude of a dot b 526 00:25:12,510 --> 00:25:16,950 is equal to the magnitude of a times the magnitude of b. 527 00:25:16,950 --> 00:25:18,450 You see, what I'm really saying here 528 00:25:18,450 --> 00:25:25,320 is, how was a dot b defined? a dot b, by definition, was what? 529 00:25:25,320 --> 00:25:29,110 It was the magnitude of a times the magnitude of b, 530 00:25:29,110 --> 00:25:32,540 times the cosine of the angle between a and b. 531 00:25:32,540 --> 00:25:36,709 Notice that if I take magnitudes here-- 532 00:25:36,709 --> 00:25:39,250 I want to keep this separate, so you can see what I'm talking 533 00:25:39,250 --> 00:25:41,300 about here-- 534 00:25:41,300 --> 00:25:41,800 Lookit. 535 00:25:41,800 --> 00:25:44,326 This factor here can be no bigger than 1. 536 00:25:44,326 --> 00:25:46,200 But in general, it's going to be less than 1. 537 00:25:46,200 --> 00:25:48,950 In other words, notice that the magnitude of a dot b 538 00:25:48,950 --> 00:25:52,410 is actually less than or equal to the magnitude of a times 539 00:25:52,410 --> 00:25:53,520 the magnitude of b. 540 00:25:53,520 --> 00:25:57,800 In fact, its equality holds only if this is a 0 degree 541 00:25:57,800 --> 00:25:59,960 angle or 180 degree angle. 542 00:25:59,960 --> 00:26:02,320 Because only then is the magnitude of the cosine 543 00:26:02,320 --> 00:26:03,560 equal to 1. 544 00:26:03,560 --> 00:26:06,057 Similar result holds for the cross product. 545 00:26:06,057 --> 00:26:07,640 The interesting point is-- and I'm not 546 00:26:07,640 --> 00:26:09,181 going to take the time to do it here, 547 00:26:09,181 --> 00:26:12,050 because I want this lecture to be basically an overview-- 548 00:26:12,050 --> 00:26:16,190 but the important point is that you don't need equality 549 00:26:16,190 --> 00:26:17,370 to prove our limit theorems. 550 00:26:17,370 --> 00:26:19,190 Remember, every one of our limit theorems 551 00:26:19,190 --> 00:26:21,470 was a string of inequalities. 552 00:26:21,470 --> 00:26:23,080 And actually-- and as I say, I'm going 553 00:26:23,080 --> 00:26:24,980 to do this in more detail in the notes-- 554 00:26:24,980 --> 00:26:28,460 the only result that we needed to prove theorems 555 00:26:28,460 --> 00:26:32,190 even in the part one section of our course 556 00:26:32,190 --> 00:26:36,100 was the fact that the less than or equal part be valid. 557 00:26:36,100 --> 00:26:38,830 The fact that it was equal when we were dealing with numbers 558 00:26:38,830 --> 00:26:40,450 was like frosting on the cake. 559 00:26:40,450 --> 00:26:43,910 We didn't need that strong a condition. 560 00:26:43,910 --> 00:26:46,190 The key overview that I want you to get 561 00:26:46,190 --> 00:26:48,320 from this present discussion is that, when 562 00:26:48,320 --> 00:26:51,750 we vectorize our definition of limit, 563 00:26:51,750 --> 00:26:55,680 that that new definition, having the same structure 564 00:26:55,680 --> 00:26:58,840 as the old-- meaning that all the properties of magnitudes 565 00:26:58,840 --> 00:27:01,590 of vectors that we need are carried over 566 00:27:01,590 --> 00:27:04,870 from absolute values of numbers, all previous limit 567 00:27:04,870 --> 00:27:07,550 theorems-- what do I mean by "all previous limit theorems"? 568 00:27:07,550 --> 00:27:11,080 I mean all the limit theorems of part one of this course 569 00:27:11,080 --> 00:27:13,330 are still valid. 570 00:27:13,330 --> 00:27:16,300 And with that in mind, I can now proceed 571 00:27:16,300 --> 00:27:18,600 to differential calculus. 572 00:27:18,600 --> 00:27:19,559 Namely what? 573 00:27:19,559 --> 00:27:21,100 I'm going to do the same thing again. 574 00:27:21,100 --> 00:27:24,410 I write down the definition of derivative as it 575 00:27:24,410 --> 00:27:26,710 was in part one of our course. 576 00:27:26,710 --> 00:27:31,440 And now what I do is I just vectorize everything in sight, 577 00:27:31,440 --> 00:27:33,390 provided it's supposed to be vectorized. 578 00:27:33,390 --> 00:27:36,370 Remember x and delta x are scalars here. 579 00:27:36,370 --> 00:27:38,590 f is the function. 580 00:27:38,590 --> 00:27:40,720 I just vectorize our definition. 581 00:27:40,720 --> 00:27:42,870 The first thing I have to do is to check 582 00:27:42,870 --> 00:27:45,350 to see whether the new expression makes sense. 583 00:27:45,350 --> 00:27:47,970 Notice that the numerator of my bracketed expression 584 00:27:47,970 --> 00:27:49,240 is now a vector. 585 00:27:49,240 --> 00:27:51,180 The denominator is a scalar. 586 00:27:51,180 --> 00:27:54,520 And a vector divided by a scalar makes perfectly good sense. 587 00:27:54,520 --> 00:27:57,590 In other words, this can be read as what? 588 00:27:57,590 --> 00:28:01,605 The scalar multiple 1 over delta x times the vector 589 00:28:01,605 --> 00:28:04,930 f of x plus delta x minus f of x. 590 00:28:04,930 --> 00:28:07,570 So this definition makes very good sense. 591 00:28:07,570 --> 00:28:10,676 It captures the meaning of average rate of change 592 00:28:10,676 --> 00:28:11,800 because you see, it's what? 593 00:28:11,800 --> 00:28:14,950 It's the total change in f over the change 594 00:28:14,950 --> 00:28:16,720 in x equal to delta x. 595 00:28:16,720 --> 00:28:18,430 So it's an average rate of change. 596 00:28:18,430 --> 00:28:21,000 Again, that's discussed in more detail in the text 597 00:28:21,000 --> 00:28:23,030 and in our notes, and in the exercises. 598 00:28:23,030 --> 00:28:26,460 But the point is that since every derivative property 599 00:28:26,460 --> 00:28:28,980 that we had in part one of our course 600 00:28:28,980 --> 00:28:31,700 followed from our limit properties, 601 00:28:31,700 --> 00:28:33,540 the fact that the limit properties are 602 00:28:33,540 --> 00:28:37,440 the same for vectors as they are for scalars now 603 00:28:37,440 --> 00:28:41,850 means that all derivative formulas are still valid. 604 00:28:41,850 --> 00:28:45,157 In other words, the product rule is still going to hold. 605 00:28:45,157 --> 00:28:46,740 The derivative of a sum is still going 606 00:28:46,740 --> 00:28:48,410 to be the sum of the derivatives. 607 00:28:48,410 --> 00:28:51,010 The derivative of a constant is still going to be 0. 608 00:28:51,010 --> 00:28:52,060 And keep this in mind. 609 00:28:52,060 --> 00:28:53,200 It's very important. 610 00:28:53,200 --> 00:28:56,460 I could re-derive every single one of these results 611 00:28:56,460 --> 00:28:59,650 from scratch as if scalar calculus had never 612 00:28:59,650 --> 00:29:04,820 been invented, just by using my basic epsilon-delta definition. 613 00:29:04,820 --> 00:29:08,190 The beauty of structure is, is that since the structures are 614 00:29:08,190 --> 00:29:10,570 alike-- you see, since they are played 615 00:29:10,570 --> 00:29:15,200 by the same rules of the game-- the theorems of vector calculus 616 00:29:15,200 --> 00:29:18,545 are going to be precisely those of scalar calculus. 617 00:29:18,545 --> 00:29:22,370 And I don't have to take the time to re-derive them all. 618 00:29:22,370 --> 00:29:24,740 As I say in the notes, I re-derive a few 619 00:29:24,740 --> 00:29:27,690 just so that you can get an idea of how this transliteration 620 00:29:27,690 --> 00:29:28,760 takes place. 621 00:29:28,760 --> 00:29:32,480 At any rate, let me close today's lesson with an example. 622 00:29:32,480 --> 00:29:35,290 Let's take motion in a plane. 623 00:29:35,290 --> 00:29:38,810 Let's suppose we have a particle moving along a curve C. 624 00:29:38,810 --> 00:29:40,540 And that the motion of the particle 625 00:29:40,540 --> 00:29:43,930 is given in parametric form, meaning we know both the x- 626 00:29:43,930 --> 00:29:47,870 and the y-coordinates of the particle at any time t, 627 00:29:47,870 --> 00:29:49,890 as functions of t. 628 00:29:49,890 --> 00:29:53,110 Notice, by the way, that what requires 629 00:29:53,110 --> 00:29:57,620 two equations in scalar form can be written as a single vector 630 00:29:57,620 --> 00:30:01,880 equation, namely, at time t-- let's say at time t, 631 00:30:01,880 --> 00:30:03,790 the particle was over here. 632 00:30:03,790 --> 00:30:08,440 Notice that if we now let r be the vector from the origin 633 00:30:08,440 --> 00:30:13,200 to the position of the particle that r is a vector, right? 634 00:30:13,200 --> 00:30:16,530 Because to specify r, I not only have to give its length, 635 00:30:16,530 --> 00:30:17,970 I have to give its direction. 636 00:30:17,970 --> 00:30:22,601 r is a vector which varies with our scalar t. 637 00:30:22,601 --> 00:30:23,100 All right? 638 00:30:23,100 --> 00:30:26,400 So r is a vector function of the scalar t. 639 00:30:26,400 --> 00:30:29,880 Now what is the i component of r? 640 00:30:29,880 --> 00:30:32,710 Well, the i component is x. 641 00:30:32,710 --> 00:30:34,920 And the j component is y. 642 00:30:34,920 --> 00:30:40,160 Notice that the pair of simultaneous parametric scalar 643 00:30:40,160 --> 00:30:43,820 equations x equals x of t, y equals y of t 644 00:30:43,820 --> 00:30:48,230 can be written as the single vector equation r of t 645 00:30:48,230 --> 00:30:54,040 equals x of t i plus y of t j, where x of t and y of t 646 00:30:54,040 --> 00:30:57,580 are scalar functions of the scalar variable t. 647 00:30:57,580 --> 00:30:59,690 OK? 648 00:30:59,690 --> 00:31:01,820 Now the question comes up, wouldn't it 649 00:31:01,820 --> 00:31:04,380 be nice to just take dr/dt here? 650 00:31:04,380 --> 00:31:08,620 Well, I mean, that's about as motivated as you can get. 651 00:31:08,620 --> 00:31:09,180 Meaning what? 652 00:31:09,180 --> 00:31:10,138 Let's see what happens. 653 00:31:10,138 --> 00:31:11,550 I know how to differentiate now. 654 00:31:11,550 --> 00:31:14,500 I'm going to use the fact that the derivative of a product 655 00:31:14,500 --> 00:31:16,530 of a sum is a sum of the derivatives, 656 00:31:16,530 --> 00:31:18,290 that i and j are constants. 657 00:31:18,290 --> 00:31:21,310 And a constant times a function, to differentiate that, 658 00:31:21,310 --> 00:31:24,260 you skip over the constant and differentiate the function. 659 00:31:24,260 --> 00:31:27,320 x of t and y of t are scalar functions, 660 00:31:27,320 --> 00:31:30,050 and I already know how to differentiate scalar functions. 661 00:31:30,050 --> 00:31:34,030 So all I'm going to assume now is that x of t and y of t 662 00:31:34,030 --> 00:31:37,420 are differentiable scalar functions, which means now 663 00:31:37,420 --> 00:31:39,830 that instead of just talking about motion in a plane, 664 00:31:39,830 --> 00:31:41,980 I'm assuming that the motion is smooth. 665 00:31:41,980 --> 00:31:45,380 At any rate, I now differentiate term by term. 666 00:31:45,380 --> 00:31:46,769 And look what I get. 667 00:31:46,769 --> 00:31:47,310 This is what? 668 00:31:47,310 --> 00:31:51,730 It's dx/dt times i plus dy/dt times j. 669 00:31:51,730 --> 00:31:55,120 In other words, the dr/dt is this particular vector. 670 00:31:55,120 --> 00:31:57,850 And this particular vector is fascinating. 671 00:31:57,850 --> 00:31:59,380 Why is it fascinating? 672 00:31:59,380 --> 00:32:02,700 Well, for one thing, let's compute its magnitude. 673 00:32:02,700 --> 00:32:06,300 The magnitude of any vector in i and j components is, what? 674 00:32:06,300 --> 00:32:09,510 The square root of the sum of the squares of the components. 675 00:32:09,510 --> 00:32:11,010 In this case, that's the square root 676 00:32:11,010 --> 00:32:16,350 of dx/dt squared plus dy/dt squared. 677 00:32:16,350 --> 00:32:18,360 But remember from part one of our course, 678 00:32:18,360 --> 00:32:23,250 this is exactly the magnitude of ds/dt where s is arc length. 679 00:32:23,250 --> 00:32:25,900 Remember I put the absolute value sign in here 680 00:32:25,900 --> 00:32:29,250 because we always take the positive square root. 681 00:32:29,250 --> 00:32:30,670 We're assuming that arc length is 682 00:32:30,670 --> 00:32:35,290 traversed in a given direction at a particular time t. 683 00:32:35,290 --> 00:32:36,920 But what is ds/dt? 684 00:32:36,920 --> 00:32:40,020 It's the change in arc length with respect to time. 685 00:32:40,020 --> 00:32:43,860 And that's precisely what we mean by speed along the curve. 686 00:32:43,860 --> 00:32:48,210 On the other hand, if we look at the slope of dr/dt, it's what? 687 00:32:48,210 --> 00:32:50,230 It's the slope-- it's determined by what? 688 00:32:50,230 --> 00:32:53,180 You take the j component, which is dy/dt, 689 00:32:53,180 --> 00:32:56,480 divided by the i component, which is dx/dt. 690 00:32:56,480 --> 00:32:59,610 By the chain rule, we know that that's dy/dx. 691 00:32:59,610 --> 00:33:03,400 Therefore, the vector dr/dt has its magnitude 692 00:33:03,400 --> 00:33:05,640 equal to the speed along the curve. 693 00:33:05,640 --> 00:33:08,480 And its direction is tangential to the curve. 694 00:33:08,480 --> 00:33:10,800 And what better motivation than that 695 00:33:10,800 --> 00:33:13,280 to define a velocity vector? 696 00:33:13,280 --> 00:33:15,670 After all, we've already done that in elementary physics. 697 00:33:15,670 --> 00:33:18,130 In elementary physics, what was the velocity vector 698 00:33:18,130 --> 00:33:19,320 defined to be? 699 00:33:19,320 --> 00:33:21,440 At a given point, it was the vector 700 00:33:21,440 --> 00:33:24,040 whose direction to the curve-- whose direction 701 00:33:24,040 --> 00:33:26,670 was tangential to the path at that point 702 00:33:26,670 --> 00:33:29,120 and whose magnitude was numerically 703 00:33:29,120 --> 00:33:31,150 equal to the speed along the curve 704 00:33:31,150 --> 00:33:32,860 that the particle had at that point. 705 00:33:32,860 --> 00:33:35,340 And that's precisely what dr/dt has. 706 00:33:35,340 --> 00:33:37,090 In other words, what we have done 707 00:33:37,090 --> 00:33:40,890 is given a self-contained mathematical derivation 708 00:33:40,890 --> 00:33:44,260 as to why the derivative of the position vector r, 709 00:33:44,260 --> 00:33:46,690 with respect to time, should physically 710 00:33:46,690 --> 00:33:49,060 be called the velocity vector. 711 00:33:49,060 --> 00:33:52,260 And then again, analogously to ordinary physics, 712 00:33:52,260 --> 00:33:56,320 if v is the velocity vector, we define the acceleration vector 713 00:33:56,320 --> 00:33:58,910 to be the derivative of the velocity vector with respect 714 00:33:58,910 --> 00:33:59,860 to time. 715 00:33:59,860 --> 00:34:01,920 Now I was originally going to close over here, 716 00:34:01,920 --> 00:34:03,570 but my feeling is that this seems 717 00:34:03,570 --> 00:34:05,580 a little bit abstract to you, so maybe we 718 00:34:05,580 --> 00:34:07,070 should take a couple more minutes 719 00:34:07,070 --> 00:34:09,670 and do a specific problem. 720 00:34:09,670 --> 00:34:12,960 Let's take the particle that moves along the path whose 721 00:34:12,960 --> 00:34:16,850 parametric equations are y equals t cubed plus 1, 722 00:34:16,850 --> 00:34:18,570 and x equals t squared. 723 00:34:18,570 --> 00:34:21,595 In other words, notice that for a given value of t-- t 724 00:34:21,595 --> 00:34:24,000 is a scalar-- for a given value of t, 725 00:34:24,000 --> 00:34:26,090 I can figure out where the particle is 726 00:34:26,090 --> 00:34:27,750 at any time along the curve. 727 00:34:27,750 --> 00:34:34,690 For example, when the time is 2, when t is 2, x is 4, y is 9. 728 00:34:34,690 --> 00:34:37,710 So at t equals 2, the particle is at the point 4 comma 729 00:34:37,710 --> 00:34:39,520 9, et cetera. 730 00:34:39,520 --> 00:34:41,219 Notice that to understand this problem, 731 00:34:41,219 --> 00:34:42,900 one does not need vectors. 732 00:34:42,900 --> 00:34:48,560 But if one knows vectors, one introduces the radius vector r. 733 00:34:48,560 --> 00:34:49,949 What is the radius vector? 734 00:34:49,949 --> 00:34:52,880 Its i component is the value of x, 735 00:34:52,880 --> 00:34:55,480 and its j component is the value of y. 736 00:34:55,480 --> 00:34:58,470 So the radius vector r is t squared i 737 00:34:58,470 --> 00:35:00,980 plus t cubed plus 1 j. 738 00:35:00,980 --> 00:35:04,230 Now, by these results, I can very quickly 739 00:35:04,230 --> 00:35:07,195 compute both the velocity and the acceleration of my particle 740 00:35:07,195 --> 00:35:08,410 at any time. 741 00:35:08,410 --> 00:35:10,670 Namely, to find the velocity, I just 742 00:35:10,670 --> 00:35:12,621 differentiate this with respect to t. 743 00:35:12,621 --> 00:35:13,870 This is going to give me what? 744 00:35:13,870 --> 00:35:16,130 The derivative of t squared is 2t. 745 00:35:16,130 --> 00:35:20,730 The derivative of t cubed plus 1 is 3 t squared. 746 00:35:20,730 --> 00:35:26,460 So my velocity vector is just 2t*i plus 3 t squared j. 747 00:35:26,460 --> 00:35:28,500 Now to get the acceleration vector, 748 00:35:28,500 --> 00:35:30,940 I simply have to differentiate the velocity vector 749 00:35:30,940 --> 00:35:32,980 with respect to time. 750 00:35:32,980 --> 00:35:34,500 And I get what? 751 00:35:34,500 --> 00:35:37,130 2i plus 6t*j. 752 00:35:37,130 --> 00:35:40,200 And I now have the acceleration, the velocity, 753 00:35:40,200 --> 00:35:44,195 and, by the way, the position of my particle at any time t. 754 00:35:44,195 --> 00:35:45,570 In particular, since I've already 755 00:35:45,570 --> 00:35:47,525 computed that the particle is at the point 756 00:35:47,525 --> 00:35:50,500 4 comma 9 when t is 2, let's carry 757 00:35:50,500 --> 00:35:53,370 through the rest of our investigation when t is 2. 758 00:35:53,370 --> 00:35:56,980 When t is 2, notice that in vector form, 759 00:35:56,980 --> 00:36:05,280 r is equal to 4i plus 9j; v is equal to 4i plus 12j; 760 00:36:05,280 --> 00:36:12,100 and a is equal to 2i plus 12j. 761 00:36:12,100 --> 00:36:15,030 Pictorially-- and by the way, to show you how to get this, 762 00:36:15,030 --> 00:36:17,630 all I'm saying here is that if we come back 763 00:36:17,630 --> 00:36:22,900 here recalling that t cubed is t squared to the 3/2 power-- 764 00:36:22,900 --> 00:36:27,010 this simply says that y is equal to x to the 3/2 plus 1. 765 00:36:27,010 --> 00:36:28,756 So I drew the path in just so that you 766 00:36:28,756 --> 00:36:30,890 can get an idea of what's going on over here. 767 00:36:30,890 --> 00:36:32,690 All I'm saying is when somebody says, 768 00:36:32,690 --> 00:36:36,090 where is the particle at time t equals 2, it's at the point 769 00:36:36,090 --> 00:36:37,620 4 comma 9. 770 00:36:37,620 --> 00:36:39,320 And notice the correlation again. 771 00:36:39,320 --> 00:36:43,650 The point 4 comma 9 is the point at which the vector 4i 772 00:36:43,650 --> 00:36:45,700 plus 9j terminates. 773 00:36:45,700 --> 00:36:49,390 Because you see, that vector originates at (0, 0). 774 00:36:49,390 --> 00:36:52,630 I can compute the velocity vector-- namely what? 775 00:36:52,630 --> 00:36:57,070 The velocity vector has its slope equal to 3-- 12 over 4. 776 00:36:57,070 --> 00:36:59,280 And its magnitude is the square root of 4 777 00:36:59,280 --> 00:37:01,080 squared plus 12 squared. 778 00:37:01,080 --> 00:37:04,380 That's the square root of 160, which is roughly 12 and a half. 779 00:37:04,380 --> 00:37:07,300 I can now draw that velocity vector to scale. 780 00:37:07,300 --> 00:37:10,530 In a similar way, the slope of the acceleration vector 781 00:37:10,530 --> 00:37:13,740 at that point is 12 over 2, which is 6. 782 00:37:13,740 --> 00:37:15,640 So the slope is 6. 783 00:37:15,640 --> 00:37:17,720 That's a pretty steep line here. 784 00:37:17,720 --> 00:37:19,110 And the magnitude is what? 785 00:37:19,110 --> 00:37:22,590 The square root of 4 plus 144-- the square root 786 00:37:22,590 --> 00:37:25,400 of 148, which I just call approximately 12, 787 00:37:25,400 --> 00:37:26,730 to give you a rough idea. 788 00:37:26,730 --> 00:37:29,260 I can draw this into scale as well. 789 00:37:29,260 --> 00:37:33,950 In other words, in just one short overview lesson, 790 00:37:33,950 --> 00:37:36,210 notice that we have introduced what 791 00:37:36,210 --> 00:37:39,440 we mean by vector functions, what we mean by limits. 792 00:37:39,440 --> 00:37:42,850 We've inherited the entire structure of part one. 793 00:37:42,850 --> 00:37:45,500 I can now introduce a position vector, 794 00:37:45,500 --> 00:37:47,320 differentiate it with respect to time 795 00:37:47,320 --> 00:37:49,640 to find velocity and acceleration, 796 00:37:49,640 --> 00:37:52,620 and I'm in business, being able to solve 797 00:37:52,620 --> 00:37:55,270 vector problems in the plane-- kinematics problems, 798 00:37:55,270 --> 00:37:57,270 if you will. 799 00:37:57,270 --> 00:38:00,160 We're going to continue on in this vein for the remainder 800 00:38:00,160 --> 00:38:00,820 of this block. 801 00:38:00,820 --> 00:38:03,300 We have other coordinate systems to talk about. 802 00:38:03,300 --> 00:38:05,500 Next time, I'm going to talk about 803 00:38:05,500 --> 00:38:09,690 tangential and normal components of vectors and the like. 804 00:38:09,690 --> 00:38:12,860 But all the time, the theory that we're talking about 805 00:38:12,860 --> 00:38:13,940 is the same. 806 00:38:13,940 --> 00:38:17,470 Namely, we are given motion in a plane. 807 00:38:17,470 --> 00:38:20,250 And we can now, in terms of vector calculus, 808 00:38:20,250 --> 00:38:23,570 study that motion very, very effectively. 809 00:38:23,570 --> 00:38:25,840 At any rate, until next time, goodbye. 810 00:38:29,230 --> 00:38:31,600 Funding for the publication of this video 811 00:38:31,600 --> 00:38:36,480 was provided by the Gabriella and Paul Rosenbaum Foundation. 812 00:38:36,480 --> 00:38:40,650 Help OCW continue to provide free and open access to MIT 813 00:38:40,650 --> 00:38:45,070 courses by making a donation at ocw.mit.edu/donate.