1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:19,500 at ocw.mit.edu. 8 00:00:34,560 --> 00:00:35,860 PROFESSOR: Hi. 9 00:00:35,860 --> 00:00:39,090 Recall that in our previous two lectures, 10 00:00:39,090 --> 00:00:43,320 we have been discussing the concepts of velocity 11 00:00:43,320 --> 00:00:47,290 and acceleration of particles moving along a curve. 12 00:00:47,290 --> 00:00:49,120 And we had pointed out originally 13 00:00:49,120 --> 00:00:53,356 that the most natural definition was in terms of say, i and j, 14 00:00:53,356 --> 00:00:55,260 or i, j, and k components-- in other words, 15 00:00:55,260 --> 00:00:57,020 Cartesian coordinates. 16 00:00:57,020 --> 00:00:59,750 And then we showed in the last lecture 17 00:00:59,750 --> 00:01:03,800 that tangential and normal components 18 00:01:03,800 --> 00:01:08,540 form a rather important system of vectors. 19 00:01:08,540 --> 00:01:10,160 And now, what we would like to do 20 00:01:10,160 --> 00:01:15,740 is to motivate the use of polar coordinate types of vectors. 21 00:01:15,740 --> 00:01:18,140 And again, it's interesting point out 22 00:01:18,140 --> 00:01:20,330 that we could have introduced polar coordinates 23 00:01:20,330 --> 00:01:23,460 into our course at any one of a number of times. 24 00:01:23,460 --> 00:01:25,690 In fact, one of the problems with the traditional 25 00:01:25,690 --> 00:01:28,560 mathematics curriculum was that, frequently, topics 26 00:01:28,560 --> 00:01:32,230 were put in just for the sake of filling a chapter. 27 00:01:32,230 --> 00:01:34,180 They fit into the space. 28 00:01:34,180 --> 00:01:36,530 They didn't need any additional prerequisites, 29 00:01:36,530 --> 00:01:37,990 so we put them in there. 30 00:01:37,990 --> 00:01:41,160 And many a typical traditional high school algebra 31 00:01:41,160 --> 00:01:43,050 book could have had the chapters shuffled 32 00:01:43,050 --> 00:01:45,170 with no loss of continuity. 33 00:01:45,170 --> 00:01:48,070 What we're trying to do in this particular course 34 00:01:48,070 --> 00:01:51,460 is to motivate why particular concepts would 35 00:01:51,460 --> 00:01:53,960 have been invented if they had not already 36 00:01:53,960 --> 00:01:55,920 have been invented previously. 37 00:01:55,920 --> 00:02:00,530 In particular, in dealing with particles moving in a plane 38 00:02:00,530 --> 00:02:03,240 or in space, there comes a time when 39 00:02:03,240 --> 00:02:06,660 one considers the very important physical application known 40 00:02:06,660 --> 00:02:09,032 as a central force field. 41 00:02:09,032 --> 00:02:10,990 Now, in a central force field-- and by the way, 42 00:02:10,990 --> 00:02:13,540 this is where the word polar coordinates comes up very 43 00:02:13,540 --> 00:02:15,010 naturally, and why we call today's 44 00:02:15,010 --> 00:02:16,450 lecture "Polar Coordinates." 45 00:02:16,450 --> 00:02:18,250 See, in a central force field, what we mean 46 00:02:18,250 --> 00:02:20,950 is that the only force acting happens 47 00:02:20,950 --> 00:02:24,230 to be located at a fixed position. 48 00:02:24,230 --> 00:02:25,870 And so that the force of attraction, 49 00:02:25,870 --> 00:02:29,780 no matter where the particle is-- wherever that particle is, 50 00:02:29,780 --> 00:02:34,010 the force is always directed along the line, 51 00:02:34,010 --> 00:02:36,350 either towards or away, depending upon whether it's 52 00:02:36,350 --> 00:02:38,950 attraction or repulsion. 53 00:02:41,740 --> 00:02:42,940 It's always acting this way. 54 00:02:42,940 --> 00:02:45,580 If we're using Newtonian mechanics, 55 00:02:45,580 --> 00:02:48,350 since the force is proportional to the acceleration, 56 00:02:48,350 --> 00:02:49,270 it says what? 57 00:02:49,270 --> 00:02:52,290 That a very natural direction for acceleration 58 00:02:52,290 --> 00:02:55,550 would be along the direction that 59 00:02:55,550 --> 00:02:58,320 connects the origin-- meaning the location 60 00:02:58,320 --> 00:03:01,710 of the central force-- to the particle. 61 00:03:01,710 --> 00:03:03,650 Well, you see, once this is done, 62 00:03:03,650 --> 00:03:06,320 it becomes a very natural thing to say, OK, 63 00:03:06,320 --> 00:03:08,970 if you're going to be talking about central force fields, 64 00:03:08,970 --> 00:03:11,720 why not introduce new variables? 65 00:03:11,720 --> 00:03:13,790 After all, it seems that the important thing now, 66 00:03:13,790 --> 00:03:15,510 in measuring the central force, is 67 00:03:15,510 --> 00:03:18,780 to know how far the particle is from the force 68 00:03:18,780 --> 00:03:22,130 and also, I suppose, to locate the position of the particle. 69 00:03:22,130 --> 00:03:23,890 Once you know how far away it is, 70 00:03:23,890 --> 00:03:28,480 you would like to know what angle the radius vector makes 71 00:03:28,480 --> 00:03:32,070 with the positive x-axis. 72 00:03:32,070 --> 00:03:34,530 And in this sense, that would be all 73 00:03:34,530 --> 00:03:37,740 that we would need to motivate polar coordinates. 74 00:03:37,740 --> 00:03:40,370 Now, you see, once the motivation is there, 75 00:03:40,370 --> 00:03:43,327 we can always study the subject independently of 76 00:03:43,327 --> 00:03:45,410 whether the motivation had ever been given or not. 77 00:03:45,410 --> 00:03:46,826 In other words, now, we say, look, 78 00:03:46,826 --> 00:03:49,420 it's important to study polar coordinates. 79 00:03:49,420 --> 00:03:50,260 Why is it important? 80 00:03:50,260 --> 00:03:52,860 Because you want to study central force fields. 81 00:03:52,860 --> 00:03:54,520 But the structure of polar coordinates 82 00:03:54,520 --> 00:03:56,950 is going to be the same, whether we study central force 83 00:03:56,950 --> 00:03:58,100 fields or not. 84 00:03:58,100 --> 00:04:03,810 Then, you see, once we finish our study of polar coordinates, 85 00:04:03,810 --> 00:04:06,210 then we say, OK, now, let's go back, 86 00:04:06,210 --> 00:04:09,385 as a particular application, to central force fields. 87 00:04:09,385 --> 00:04:11,500 At any rate, without further ado, 88 00:04:11,500 --> 00:04:14,690 let's tackle the subject of polar coordinates. 89 00:04:14,690 --> 00:04:18,839 Again, many of you may be very, very familiar 90 00:04:18,839 --> 00:04:21,470 with polar coordinates, some of you a bit rusty. 91 00:04:21,470 --> 00:04:26,630 To play it safe, I have divided the study guide in such a way 92 00:04:26,630 --> 00:04:30,340 that there will be two units devoted to polar coordinates, 93 00:04:30,340 --> 00:04:33,940 but that this one lecture will cover both of those two units, 94 00:04:33,940 --> 00:04:36,737 that this lecture, as usual, will be the overview. 95 00:04:36,737 --> 00:04:39,070 And I guess this is the best you can do with mathematics 96 00:04:39,070 --> 00:04:40,140 beyond a point. 97 00:04:40,140 --> 00:04:42,120 Beyond a certain point, mathematics 98 00:04:42,120 --> 00:04:44,830 ceases to be a spectator sport, and you just 99 00:04:44,830 --> 00:04:47,110 have to dirty your hands with the computations, 100 00:04:47,110 --> 00:04:50,520 and that the basic theory, as simple as it may seem, 101 00:04:50,520 --> 00:04:51,950 is straightforward. 102 00:04:51,950 --> 00:04:54,870 It's just a case of picking up the computational know-how 103 00:04:54,870 --> 00:04:57,390 together with a familiarity so that you 104 00:04:57,390 --> 00:04:59,150 feel at ease with the concepts. 105 00:04:59,150 --> 00:05:01,460 So let's just quickly go through the highlights 106 00:05:01,460 --> 00:05:02,830 of polar coordinates. 107 00:05:02,830 --> 00:05:06,270 They start off in a very deceptively simple way. 108 00:05:06,270 --> 00:05:08,580 Namely, given a point P in the plane, 109 00:05:08,580 --> 00:05:11,210 if we elect to use Cartesian coordinates, 110 00:05:11,210 --> 00:05:15,220 the point P can be labeled as x comma y. 111 00:05:15,220 --> 00:05:19,080 On the other hand, using polar coordinates-- referring 112 00:05:19,080 --> 00:05:21,760 to this diagram, at least-- one would 113 00:05:21,760 --> 00:05:25,440 tend, as we mentioned before, to label this point r comma theta, 114 00:05:25,440 --> 00:05:28,490 where r is the distance of the point from the origin, 115 00:05:28,490 --> 00:05:31,900 and theta the angle that we're talking about over here. 116 00:05:31,900 --> 00:05:34,740 Notice, by the way, that there is a very simple set 117 00:05:34,740 --> 00:05:37,230 of formulae by which we can switch 118 00:05:37,230 --> 00:05:40,230 from one pair of representations to the other. 119 00:05:40,230 --> 00:05:44,640 For example, to express x and y in terms of r and theta, 120 00:05:44,640 --> 00:05:47,450 one simply has that x equals r cosine theta, 121 00:05:47,450 --> 00:05:50,080 y equals r sine theta. 122 00:05:50,080 --> 00:05:50,770 OK. 123 00:05:50,770 --> 00:05:54,940 And conversely, to express r and theta in terms of x and y, 124 00:05:54,940 --> 00:05:57,820 one has that r squared equals x squared 125 00:05:57,820 --> 00:06:01,504 plus y squared and the tan theta is y over x. 126 00:06:01,504 --> 00:06:02,920 By the way, you might wonder why I 127 00:06:02,920 --> 00:06:05,160 didn't write r equals the square root of x 128 00:06:05,160 --> 00:06:06,930 squared plus y squared. 129 00:06:06,930 --> 00:06:08,580 The reason I wrote this is that we're 130 00:06:08,580 --> 00:06:11,760 going to get into some complications very shortly, 131 00:06:11,760 --> 00:06:14,440 complications which I shall skim over. 132 00:06:14,440 --> 00:06:15,940 And if you see how long-winded I am, 133 00:06:15,940 --> 00:06:18,745 skimming over is not going to be that short, but short relative 134 00:06:18,745 --> 00:06:20,120 to the treatment that we're going 135 00:06:20,120 --> 00:06:22,060 to give it in the exercises. 136 00:06:22,060 --> 00:06:23,950 There are certain complications that set in, 137 00:06:23,950 --> 00:06:26,356 because r can be negative. 138 00:06:26,356 --> 00:06:27,730 By the same token, somebody says, 139 00:06:27,730 --> 00:06:29,550 instead of writing tan theta equals y 140 00:06:29,550 --> 00:06:32,050 over x, why couldn't you say that theta 141 00:06:32,050 --> 00:06:36,330 was the inverse tangent y over x, use the inverse trig 142 00:06:36,330 --> 00:06:37,260 functions? 143 00:06:37,260 --> 00:06:40,400 Remember that the inverse trig functions require 144 00:06:40,400 --> 00:06:42,640 a principal value domain. 145 00:06:42,640 --> 00:06:44,020 For the tangent that's what? 146 00:06:44,020 --> 00:06:47,260 Between minus pi over 2 and pi over 2. 147 00:06:47,260 --> 00:06:49,640 And notice that the angle theta certainly is not 148 00:06:49,640 --> 00:06:50,734 restricted to that range. 149 00:06:50,734 --> 00:06:53,150 Theta could be in the second quadrant, the third quadrant. 150 00:06:53,150 --> 00:06:54,610 It can wind around. 151 00:06:54,610 --> 00:06:57,060 In other words, one of the luxuries 152 00:06:57,060 --> 00:07:01,130 about Cartesian coordinates was that they were terribly 153 00:07:01,130 --> 00:07:03,440 formally stilted. 154 00:07:03,440 --> 00:07:06,710 There were no two ways to represent the same point. 155 00:07:06,710 --> 00:07:08,840 In the study of Cartesian coordinates, 156 00:07:08,840 --> 00:07:13,030 one can say, to a point, position is everything in life, 157 00:07:13,030 --> 00:07:17,790 that once you've changed the coordinates, 158 00:07:17,790 --> 00:07:19,180 you've changed the point. 159 00:07:19,180 --> 00:07:21,760 For example, there's no possible way 160 00:07:21,760 --> 00:07:26,150 for the point x comma y to equal the point, say, 2 comma 3, 161 00:07:26,150 --> 00:07:28,810 unless x equals 2 or y equals 3. 162 00:07:28,810 --> 00:07:31,330 In other words, for Cartesian coordinates, 163 00:07:31,330 --> 00:07:34,260 the only way to points could be equal would be if they were 164 00:07:34,260 --> 00:07:36,660 equal coordinate by coordinate. 165 00:07:36,660 --> 00:07:40,010 The complication that sets in for polar coordinates 166 00:07:40,010 --> 00:07:43,840 is that, unfortunately, this simple result no longer remains 167 00:07:43,840 --> 00:07:44,470 true. 168 00:07:44,470 --> 00:07:48,530 For example, if I tell you that the point r_1 comma 169 00:07:48,530 --> 00:07:51,990 theta_1 is equal to the point r_2 comma theta_2, 170 00:07:51,990 --> 00:07:55,840 it does not imply-- that's a slash through the arrow here. 171 00:07:55,840 --> 00:07:57,100 This means it's false. 172 00:07:57,100 --> 00:08:00,770 It does not imply that I can conclude that r_1 equals r:2 173 00:08:00,770 --> 00:08:02,590 and that theta_1 equals theta_2. 174 00:08:02,590 --> 00:08:05,900 To be sure, it's possible that r_1 equals r_2 175 00:08:05,900 --> 00:08:08,510 and it's possible that theta 1 equals theta 2, 176 00:08:08,510 --> 00:08:10,660 but it doesn't have to happen. 177 00:08:10,660 --> 00:08:12,380 Well, the nice thing about mathematics 178 00:08:12,380 --> 00:08:14,190 is that we can always give examples 179 00:08:14,190 --> 00:08:15,680 to back up what we're saying. 180 00:08:15,680 --> 00:08:18,230 Let's look at a few examples. 181 00:08:18,230 --> 00:08:20,970 Let's look, for example, at what happens, 182 00:08:20,970 --> 00:08:24,510 if we're using radian measure, if we replace 183 00:08:24,510 --> 00:08:29,790 the angle by some integral multiple of 2 pi added 184 00:08:29,790 --> 00:08:30,930 on to that angle. 185 00:08:30,930 --> 00:08:33,940 In other words, let's take the point r_1 comma theta_1 186 00:08:33,940 --> 00:08:38,900 and compare that point with the point whose coordinates are r_1 187 00:08:38,900 --> 00:08:42,770 comma theta_1 plus 2k*pi. 188 00:08:42,770 --> 00:08:46,010 Look, the r values are the same here. 189 00:08:46,010 --> 00:08:50,080 But look at theta_1 and theta_1 plus 2k*pi. 190 00:08:50,080 --> 00:08:56,700 Notice that, unless k is 0, these are different angles. 191 00:08:56,700 --> 00:08:58,200 It's rather interesting. 192 00:08:58,200 --> 00:09:00,400 I know in high school this happens a lot. 193 00:09:00,400 --> 00:09:01,960 Youngsters will tend to say things 194 00:09:01,960 --> 00:09:05,940 like 30 degrees is the same as 390 degrees. 195 00:09:05,940 --> 00:09:08,730 And in a way, they're right, except that they say it in such 196 00:09:08,730 --> 00:09:10,230 a way that it's dangerous. 197 00:09:10,230 --> 00:09:12,640 Certainly, one should not confuse 30 degrees 198 00:09:12,640 --> 00:09:14,050 with 390 degrees. 199 00:09:14,050 --> 00:09:17,780 What one usually means is that any trigonometric function 200 00:09:17,780 --> 00:09:21,390 of 30 degrees is equal to that same trigonometric function 201 00:09:21,390 --> 00:09:23,240 of 390 degrees. 202 00:09:23,240 --> 00:09:25,440 In other words, the technical way of saying it 203 00:09:25,440 --> 00:09:27,330 is that the trigonometric functions 204 00:09:27,330 --> 00:09:32,190 are periodic with period 2pi in radian measure, 360 205 00:09:32,190 --> 00:09:34,350 degrees in degree measure. 206 00:09:34,350 --> 00:09:37,220 In other words, for example, using radian measure, 207 00:09:37,220 --> 00:09:41,030 notice that the sine of theta_1 is equal to the sine of theta_1 208 00:09:41,030 --> 00:09:45,140 plus 2*pi*k, where k is some integer here. 209 00:09:45,140 --> 00:09:51,200 But that theta_1 is unequal to theta_1 plus 2*pi*k if k is 210 00:09:51,200 --> 00:09:51,900 unequal to 0. 211 00:09:51,900 --> 00:09:53,440 If k is equal to 0, of course, this 212 00:09:53,440 --> 00:09:55,100 happens to be a special case. 213 00:09:55,100 --> 00:09:58,340 You see, the whole point is that all it 214 00:09:58,340 --> 00:10:00,010 means in terms of a graph, if you 215 00:10:00,010 --> 00:10:03,950 were to plot the curve y equals sine x, 216 00:10:03,950 --> 00:10:05,700 it is possible for what? 217 00:10:05,700 --> 00:10:09,160 Many different values of theta, many different values 218 00:10:09,160 --> 00:10:12,850 of x, to give the same value of sine x. 219 00:10:12,850 --> 00:10:16,000 What this means, by the way, in terms of polar coordinates, 220 00:10:16,000 --> 00:10:18,820 is that it certainly makes a difference whether we talk 221 00:10:18,820 --> 00:10:25,550 about a line making an angle of 30 222 00:10:25,550 --> 00:10:29,300 degrees with the positive x-axis or making 390 degrees, 223 00:10:29,300 --> 00:10:31,220 because, you see, that 390 degrees 224 00:10:31,220 --> 00:10:34,720 seems to indicate that you've made one full circuit and then 225 00:10:34,720 --> 00:10:36,770 an additional 30 degrees. 226 00:10:36,770 --> 00:10:39,820 As far as position is concerned, you're in the same place. 227 00:10:39,820 --> 00:10:42,490 But for example, in terms of fuel consumption, 228 00:10:42,490 --> 00:10:45,770 it uses more fuel, say, to go through the 390 degrees 229 00:10:45,770 --> 00:10:47,332 than to go through the 30 degrees. 230 00:10:47,332 --> 00:10:49,790 But at any rate, that's not the point we want to make here. 231 00:10:49,790 --> 00:10:52,140 The point is, notice in this particular example 232 00:10:52,140 --> 00:10:56,020 that this is two different names for the same point. 233 00:10:56,020 --> 00:10:59,600 But we cannot conclude that, coordinate by coordinate, 234 00:10:59,600 --> 00:11:02,240 the coordinates are equal. 235 00:11:02,240 --> 00:11:04,040 That's not the worst of it. 236 00:11:04,040 --> 00:11:07,810 The worst of it is that r and/or theta don't even 237 00:11:07,810 --> 00:11:08,860 have to be positive. 238 00:11:08,860 --> 00:11:11,210 Another way of saying that is they may be negative. 239 00:11:11,210 --> 00:11:14,760 For example, to capitalize on the idea of vector notation, 240 00:11:14,760 --> 00:11:17,160 what one frequently does is says, 241 00:11:17,160 --> 00:11:19,850 look, let's talk about a negative distance. 242 00:11:19,850 --> 00:11:22,959 And by a negative distance, we'll identify that with sense. 243 00:11:22,959 --> 00:11:25,000 In other words, let's call this angle here theta. 244 00:11:27,899 --> 00:11:29,690 See, notice, what you're really saying here 245 00:11:29,690 --> 00:11:31,680 is you're assuming that the radius vector, when 246 00:11:31,680 --> 00:11:34,880 you're calling this angle theta, has this particular sense. 247 00:11:34,880 --> 00:11:37,400 Suppose you wanted to talk about this vector. 248 00:11:37,400 --> 00:11:42,420 Notice that this would be the right sense if the angle here 249 00:11:42,420 --> 00:11:43,480 happened to be what? 250 00:11:43,480 --> 00:11:49,680 Theta plus pi-- 180 degrees, pi radians. 251 00:11:49,680 --> 00:11:51,390 Now, what you say is, if you're looking 252 00:11:51,390 --> 00:11:54,310 at this particular point, notice that with respect 253 00:11:54,310 --> 00:11:57,440 to this vector, you must go what? 254 00:11:57,440 --> 00:11:59,260 Negative r units-- in other words, 255 00:11:59,260 --> 00:12:03,050 you must move in the opposite sense of this particular vector 256 00:12:03,050 --> 00:12:04,780 to get to this point. 257 00:12:04,780 --> 00:12:08,240 And again, this is a touchy enough concept 258 00:12:08,240 --> 00:12:10,550 that we're going to do this in great detail 259 00:12:10,550 --> 00:12:12,610 in the learning exercises. 260 00:12:12,610 --> 00:12:16,560 But the point is, notice that r comma theta is 261 00:12:16,560 --> 00:12:20,410 a different name for the same point as that which is named 262 00:12:20,410 --> 00:12:23,354 by minus r comma theta plus pi. 263 00:12:23,354 --> 00:12:24,020 Do you see that? 264 00:12:24,020 --> 00:12:26,170 Let's look at that one more time just to make sure. 265 00:12:26,170 --> 00:12:30,560 See, on the one hand, reading this angle and this distance, 266 00:12:30,560 --> 00:12:33,170 this is r comma theta. 267 00:12:33,170 --> 00:12:36,590 On the other hand, reading it from this angle, 268 00:12:36,590 --> 00:12:41,324 the value is minus r and the angle is theta plus pi. 269 00:12:41,324 --> 00:12:43,490 And I'll get a different piece of chalk in a minute, 270 00:12:43,490 --> 00:12:45,531 because this doesn't seem to be writing too well. 271 00:12:45,531 --> 00:12:47,730 But let's not worry about that for the time being. 272 00:12:47,730 --> 00:12:49,010 Let me give you an example. 273 00:12:49,010 --> 00:12:51,450 Look at the curve whose polar equation is 274 00:12:51,450 --> 00:12:53,780 r equals sine squared theta. 275 00:12:53,780 --> 00:12:58,080 Notice that if theta equals pi over 6, the sine of pi over 6 276 00:12:58,080 --> 00:12:59,060 is 1/2. 277 00:12:59,060 --> 00:13:01,380 1/2 squared is 1/4. 278 00:13:01,380 --> 00:13:05,560 So notice that r equals 1/4, theta equals pi over 6 279 00:13:05,560 --> 00:13:08,410 satisfies this particular equation. 280 00:13:08,410 --> 00:13:10,750 Now, notice that another name for the same point, 281 00:13:10,750 --> 00:13:16,970 using this recipe here, is minus 1/4 comma 7*pi over 6. 282 00:13:16,970 --> 00:13:18,790 But without even looking, you should 283 00:13:18,790 --> 00:13:22,325 be able to see that it's impossible for this value of r 284 00:13:22,325 --> 00:13:25,610 and this value of theta to satisfy this equation. 285 00:13:25,610 --> 00:13:28,070 In particular, notice that sine squared 286 00:13:28,070 --> 00:13:29,480 theta can't be negative. 287 00:13:29,480 --> 00:13:32,720 Just by reading this equation, r can never be negative, 288 00:13:32,720 --> 00:13:35,030 therefore, how can r equals minus 1/4 289 00:13:35,030 --> 00:13:37,290 possibly satisfy this equation? 290 00:13:37,290 --> 00:13:39,030 And this is where the big complication 291 00:13:39,030 --> 00:13:41,620 comes in in terms of simultaneous equations 292 00:13:41,620 --> 00:13:42,620 and what have you. 293 00:13:42,620 --> 00:13:43,900 Here, we have what? 294 00:13:43,900 --> 00:13:46,830 Two different names for the same point. 295 00:13:46,830 --> 00:13:51,410 Yet, by one of its names, the point satisfies the equation, 296 00:13:51,410 --> 00:13:56,010 and by another name, it doesn't satisfy the equation. 297 00:13:56,010 --> 00:13:57,730 And this is a very, very touchy thing, 298 00:13:57,730 --> 00:13:59,210 because you'd like to believe what? 299 00:13:59,210 --> 00:14:02,410 That a point belongs to a curve, not the name of a point. 300 00:14:02,410 --> 00:14:05,330 And that one of the difficulties with polar coordinates 301 00:14:05,330 --> 00:14:07,790 is that, to check whether a point belongs to a curve 302 00:14:07,790 --> 00:14:12,430 or not, if the point, as named by one way, 303 00:14:12,430 --> 00:14:15,070 doesn't satisfy the equation, you still 304 00:14:15,070 --> 00:14:16,950 have to check other possible names 305 00:14:16,950 --> 00:14:19,700 to see whether they satisfy the equation or not. 306 00:14:19,700 --> 00:14:21,770 And this is a very complicated topic. 307 00:14:21,770 --> 00:14:24,960 And this is why some of our learning exercises in this unit 308 00:14:24,960 --> 00:14:26,100 take so many pages. 309 00:14:26,100 --> 00:14:29,680 It's more or less devoted to giving you insight into this 310 00:14:29,680 --> 00:14:31,530 if you don't already have this insight. 311 00:14:31,530 --> 00:14:37,910 But at any rate, let's continue on in general terms here. 312 00:14:37,910 --> 00:14:40,530 The next most important thing to talk about 313 00:14:40,530 --> 00:14:43,060 is to make sure that we understand what 314 00:14:43,060 --> 00:14:45,150 polar coordinates really mean. 315 00:14:45,150 --> 00:14:47,590 Namely, when someone says, I am thinking 316 00:14:47,590 --> 00:14:50,250 of the curve C whose polar equation 317 00:14:50,250 --> 00:14:57,660 is r equals sine theta, he does not mean plot r versus theta 318 00:14:57,660 --> 00:15:01,200 this way and call that the curve C. You see, 319 00:15:01,200 --> 00:15:04,370 notice that even though you're calling this r and theta, 320 00:15:04,370 --> 00:15:07,520 using the axes to be at right angles this way, 321 00:15:07,520 --> 00:15:10,050 no matter how you slice it, you're still 322 00:15:10,050 --> 00:15:12,970 using Cartesian coordinates here, 323 00:15:12,970 --> 00:15:16,390 only what theta replacing the name of x and r 324 00:15:16,390 --> 00:15:18,030 replacing the name y. 325 00:15:18,030 --> 00:15:19,606 See, this is not what's meant. 326 00:15:19,606 --> 00:15:20,980 Remember that when you're talking 327 00:15:20,980 --> 00:15:24,810 about polar coordinates, r specifically measures what? 328 00:15:24,810 --> 00:15:29,880 The distance of the particular point on C from the origin, 329 00:15:29,880 --> 00:15:32,740 and theta measures the angle that that radius vector 330 00:15:32,740 --> 00:15:34,780 makes with the x-axis. 331 00:15:34,780 --> 00:15:37,820 For example, what we do we mean by the curve whose 332 00:15:37,820 --> 00:15:41,090 polar equation is r equals sine theta 333 00:15:41,090 --> 00:15:51,260 is the circle of radius 1/2 centered on the y-axis, 334 00:15:51,260 --> 00:15:52,500 you see, at the point what? 335 00:15:52,500 --> 00:15:55,890 In Cartesian coordinates, x is 0, y is 1/2. 336 00:15:55,890 --> 00:15:58,370 And I claim that that's what we mean by the curve r 337 00:15:58,370 --> 00:15:59,530 equals sine theta. 338 00:15:59,530 --> 00:16:01,110 And how can I prove that to you? 339 00:16:01,110 --> 00:16:03,430 Well, the easiest way to prove that to you 340 00:16:03,430 --> 00:16:05,510 is by some elementary geometry. 341 00:16:05,510 --> 00:16:07,900 Namely, I call this point r comma theta. 342 00:16:07,900 --> 00:16:10,180 I now draw in these guidelines. 343 00:16:10,180 --> 00:16:13,300 Notice that, by definition, this length is r. 344 00:16:13,300 --> 00:16:16,710 Since this is an inscribed angle on a diameter, 345 00:16:16,710 --> 00:16:18,580 it's a 90-degree angle. 346 00:16:18,580 --> 00:16:22,360 Therefore, since this angle is theta and both of these angles 347 00:16:22,360 --> 00:16:24,420 are complements of a 90-degree angle, 348 00:16:24,420 --> 00:16:26,470 this angle must also be theta. 349 00:16:26,470 --> 00:16:29,360 And now, if I just look at this particular diagram, 350 00:16:29,360 --> 00:16:32,920 I read this right triangle, and immediately, I see what? 351 00:16:32,920 --> 00:16:36,520 That from this right triangle, r equals sine theta. 352 00:16:36,520 --> 00:16:39,200 In other words, the curve C, whose 353 00:16:39,200 --> 00:16:42,050 polar equation is r equals sine theta, 354 00:16:42,050 --> 00:16:43,810 is this particular curve. 355 00:16:43,810 --> 00:16:48,400 By the way, we often get indoctrinated 356 00:16:48,400 --> 00:16:51,040 into seeing things in a very natural way in terms 357 00:16:51,040 --> 00:16:52,610 of a native tongue. 358 00:16:52,610 --> 00:16:55,390 My father-in-law was brought up in Russia. 359 00:16:55,390 --> 00:16:57,200 He had a little grocery store. 360 00:16:57,200 --> 00:16:59,120 He spoke fluent English, but when 361 00:16:59,120 --> 00:17:02,360 he added up customers' bills, he always added in Russian. 362 00:17:02,360 --> 00:17:05,420 He was more at home with Russia numerals. 363 00:17:05,420 --> 00:17:08,050 And this always impressed me, until I realized 364 00:17:08,050 --> 00:17:10,569 that I was the same way, only I use 365 00:17:10,569 --> 00:17:13,859 base 10 numerals instead of base 7 or something like this. 366 00:17:13,859 --> 00:17:15,960 And the same thing happens with the polar 367 00:17:15,960 --> 00:17:17,720 versus Cartesian coordinates. 368 00:17:17,720 --> 00:17:19,270 Even though polar coordinates are 369 00:17:19,270 --> 00:17:21,660 independent of Cartesian coordinates, 370 00:17:21,660 --> 00:17:23,530 the fact remains that we're probably more 371 00:17:23,530 --> 00:17:25,640 at home with Cartesian coordinates 372 00:17:25,640 --> 00:17:27,339 than we are with polar coordinates. 373 00:17:27,339 --> 00:17:29,370 Consequently, one very common trick, 374 00:17:29,370 --> 00:17:32,560 when we can get away with it, is to take a polar equation 375 00:17:32,560 --> 00:17:35,810 and translate it into an equivalent Cartesian equation. 376 00:17:35,810 --> 00:17:38,030 Namely, given r equals sine theta, 377 00:17:38,030 --> 00:17:41,430 and remembering that r squared is x squared plus y squared 378 00:17:41,430 --> 00:17:44,330 and that r sine theta is y, we multiply 379 00:17:44,330 --> 00:17:47,410 both sides of this equation by r to get r squared 380 00:17:47,410 --> 00:17:49,040 equals r sin(theta). 381 00:17:49,040 --> 00:17:52,100 This leads to x squared plus y squared equals y. 382 00:17:52,100 --> 00:17:54,390 If we then complete the square, et cetera, 383 00:17:54,390 --> 00:17:56,770 we find that, in Cartesian coordinates, 384 00:17:56,770 --> 00:18:00,210 this is the equation of the circle centered at the point 0 385 00:18:00,210 --> 00:18:03,590 comma 1/2 with radius equal to 1/2. 386 00:18:03,590 --> 00:18:06,270 The thing that's very, very important to note here 387 00:18:06,270 --> 00:18:10,070 is that, in terms of x and y, the relationship 388 00:18:10,070 --> 00:18:14,160 x squared plus y squared equals y is not structurally 389 00:18:14,160 --> 00:18:17,960 the same as the relationship r equals sine theta. 390 00:18:17,960 --> 00:18:20,600 In other words, the curve C is the same 391 00:18:20,600 --> 00:18:22,490 whether you use its Cartesian form 392 00:18:22,490 --> 00:18:24,570 or whether you use its polar form. 393 00:18:24,570 --> 00:18:26,710 But what's very important to note 394 00:18:26,710 --> 00:18:29,950 is that the relationship between r and theta 395 00:18:29,950 --> 00:18:34,500 is not algebraically the same as the relationship between x 396 00:18:34,500 --> 00:18:36,270 and y. 397 00:18:36,270 --> 00:18:40,250 But the important point is that, since the curve 398 00:18:40,250 --> 00:18:44,580 C does not measure r versus theta as being at right angles 399 00:18:44,580 --> 00:18:48,070 to each other, that if you were given r equals sine theta 400 00:18:48,070 --> 00:18:50,250 and you compute dr / d theta, which, in this case, 401 00:18:50,250 --> 00:18:51,390 simply would be what? 402 00:18:51,390 --> 00:18:55,142 cosine theta-- that does not have the slope of C. 403 00:18:55,142 --> 00:18:57,100 You see, this is what I want you to understand. 404 00:18:57,100 --> 00:18:59,410 If I compute dr / d theta and one 405 00:18:59,410 --> 00:19:02,850 would interpret that as a slope, it's not a slope of the curve 406 00:19:02,850 --> 00:19:06,400 C. It's a slope of this particular curve which wasn't 407 00:19:06,400 --> 00:19:09,700 C. In other words, this is the curve 408 00:19:09,700 --> 00:19:15,590 that measures r versus theta in terms of Cartesian coordinates. 409 00:19:15,590 --> 00:19:17,560 That dr / d theta just shows you how 410 00:19:17,560 --> 00:19:19,860 r is changing with respect to theta, 411 00:19:19,860 --> 00:19:24,790 it does not tell you how the curve is rising at that point. 412 00:19:24,790 --> 00:19:27,600 And again, more information is given 413 00:19:27,600 --> 00:19:29,490 on this in terms of exercises. 414 00:19:29,490 --> 00:19:31,430 But that's what I do want you to understand. 415 00:19:31,430 --> 00:19:34,655 It does not mean that you can't compute the slope of this curve 416 00:19:34,655 --> 00:19:35,570 at any point. 417 00:19:35,570 --> 00:19:38,590 It does mean that if you want the slope 418 00:19:38,590 --> 00:19:42,110 and you compute it by letting it equal dr / d theta 419 00:19:42,110 --> 00:19:44,740 you will get an answer this way, but it won't be the slope 420 00:19:44,740 --> 00:19:45,900 that you're looking for. 421 00:19:45,900 --> 00:19:48,700 In fact, what I hope that you can see rather simply 422 00:19:48,700 --> 00:19:50,995 here is the following. 423 00:19:54,880 --> 00:19:56,680 Say I want the slope of the tangent line 424 00:19:56,680 --> 00:19:58,530 to the curve at this particular point, 425 00:19:58,530 --> 00:20:00,450 it doesn't really make much difference 426 00:20:00,450 --> 00:20:05,560 whether I use the angle phi-- which the tangent line makes 427 00:20:05,560 --> 00:20:09,372 with the positive x-axis-- or whether I use phi or "phi," 428 00:20:09,372 --> 00:20:14,870 or, I don't know-- This angle is called either "phi" or "phi". 429 00:20:14,870 --> 00:20:17,720 And this angle is called "psi" or "psi." 430 00:20:17,720 --> 00:20:19,470 I get all mixed up with the Greek letters. 431 00:20:19,470 --> 00:20:20,770 You know, in fact, coming into work today, 432 00:20:20,770 --> 00:20:22,145 I was listening to a disc jockey, 433 00:20:22,145 --> 00:20:24,662 and he was making fun of how in drugstores now, you 434 00:20:24,662 --> 00:20:26,370 can get a full course meal, but you can't 435 00:20:26,370 --> 00:20:28,306 get any medicines anymore. 436 00:20:28,306 --> 00:20:29,930 And he was talking about his friend who 437 00:20:29,930 --> 00:20:33,284 got all A's in pharmacy school, and they flunked him out 438 00:20:33,284 --> 00:20:35,200 because he didn't know how to make a sandwich. 439 00:20:35,200 --> 00:20:37,432 I almost flunked out of math because I have trouble 440 00:20:37,432 --> 00:20:38,640 with the Greek alphabet here. 441 00:20:38,640 --> 00:20:40,830 But look, forget about that. 442 00:20:40,830 --> 00:20:42,600 The thing I want to see is I want 443 00:20:42,600 --> 00:20:44,430 to know the direction of this line. 444 00:20:44,430 --> 00:20:48,000 If it's convenient to use phi, I'll use phi. 445 00:20:48,000 --> 00:20:51,600 If it's convenient to use psi, I'll use psi. 446 00:20:51,600 --> 00:20:52,780 What is phi? 447 00:20:52,780 --> 00:20:55,410 Phi is the angle that the tangent line 448 00:20:55,410 --> 00:20:57,110 makes with the positive x-axis. 449 00:20:57,110 --> 00:20:58,516 What is psi? 450 00:20:58,516 --> 00:21:01,090 Psi is the angle that the tangent line 451 00:21:01,090 --> 00:21:03,180 makes with the radius vector. 452 00:21:03,180 --> 00:21:06,450 Now, as I'll show you in some of our exercises, 453 00:21:06,450 --> 00:21:10,760 to use the definition that the slope is 454 00:21:10,760 --> 00:21:15,500 tan phi and translating that from Cartesian coordinates 455 00:21:15,500 --> 00:21:19,260 into polar coordinates is a very, very messy job. 456 00:21:19,260 --> 00:21:21,940 What turns out to be very, very interesting, 457 00:21:21,940 --> 00:21:25,210 at least from my point of view, is that not only is the angle 458 00:21:25,210 --> 00:21:29,050 psi-- "psi"-- more natural to use than phi when 459 00:21:29,050 --> 00:21:30,800 you're dealing with polar coordinates, 460 00:21:30,800 --> 00:21:34,870 but it almost turns out that because it was more natural, 461 00:21:34,870 --> 00:21:37,180 there's a simpler formula for it. 462 00:21:37,180 --> 00:21:41,290 In other words, whereas the formula for tan phi 463 00:21:41,290 --> 00:21:44,990 is very, very complicated in polar coordinates, 464 00:21:44,990 --> 00:21:48,120 the formula for tan psi is very simply 465 00:21:48,120 --> 00:21:51,920 given by r divided by dr / d theta. 466 00:21:51,920 --> 00:21:54,560 In other words, if I take r, divide 467 00:21:54,560 --> 00:21:58,860 that by dr / d theta, what I get is the tangent of this angle. 468 00:21:58,860 --> 00:21:59,450 Now, look. 469 00:21:59,450 --> 00:22:01,590 Once I have the tangent of this angle, 470 00:22:01,590 --> 00:22:03,466 it's very simple to construct a tangent line. 471 00:22:03,466 --> 00:22:04,715 That's what I want you to see. 472 00:22:04,715 --> 00:22:05,280 Look. 473 00:22:05,280 --> 00:22:08,570 Suppose this is the point P, and this is my pole O, 474 00:22:08,570 --> 00:22:09,870 for polar coordinates. 475 00:22:09,870 --> 00:22:13,420 And given that r was some function of theta, 476 00:22:13,420 --> 00:22:16,230 suppose now I've computed dr / d theta, 477 00:22:16,230 --> 00:22:20,780 I've divided that into r, and I've now found what tan psi is. 478 00:22:20,780 --> 00:22:23,840 What I do is I take the line of action that 479 00:22:23,840 --> 00:22:30,680 joins the origin to P, I construct the angle psi, 480 00:22:30,680 --> 00:22:33,620 draw this line, and whatever that line is, 481 00:22:33,620 --> 00:22:36,780 that is the line which is tangent to my particular curve 482 00:22:36,780 --> 00:22:38,610 at the point P. 483 00:22:38,610 --> 00:22:40,650 But the thing I want to see is that we do not 484 00:22:40,650 --> 00:22:43,550 need Cartesian coordinates at all in order 485 00:22:43,550 --> 00:22:47,530 to be able to tackle calculus properties when an equation is 486 00:22:47,530 --> 00:22:48,990 written in polar coordinates. 487 00:22:48,990 --> 00:22:51,070 But what there is a tendency to do is what? 488 00:22:51,070 --> 00:22:53,766 That when we're dealing with polar coordinates, 489 00:22:53,766 --> 00:22:55,390 since we're more at home with Cartesian 490 00:22:55,390 --> 00:22:58,360 coordinates, we will often be tempted to switch everything 491 00:22:58,360 --> 00:23:01,750 into Cartesian coordinates, which is perfectly fair game. 492 00:23:01,750 --> 00:23:03,530 But the important thing is to remember 493 00:23:03,530 --> 00:23:07,560 that all of our calculus results can be derived independently 494 00:23:07,560 --> 00:23:10,540 of whether Cartesian coordinates were ever invented. 495 00:23:10,540 --> 00:23:12,120 For example, the final concept I want 496 00:23:12,120 --> 00:23:16,500 to talk about in this lecture is how one would have studied area 497 00:23:16,500 --> 00:23:19,450 if one had only polar coordinates and had never 498 00:23:19,450 --> 00:23:22,020 had Cartesian coordinates. 499 00:23:22,020 --> 00:23:26,100 By the way, notice also that, in terms of polar coordinates, 500 00:23:26,100 --> 00:23:29,380 one is more interested in sectors 501 00:23:29,380 --> 00:23:30,810 than in rectangular grids. 502 00:23:30,810 --> 00:23:34,200 In other words, if you look at something ranging from theta_1 503 00:23:34,200 --> 00:23:36,560 to theta_2, you think of something 504 00:23:36,560 --> 00:23:40,040 caught between two rays here. 505 00:23:40,040 --> 00:23:42,920 And let's suppose I wanted the area of this particular region, 506 00:23:42,920 --> 00:23:45,480 where this particular curve happened to have the form, 507 00:23:45,480 --> 00:23:49,140 say, r equals g of theta-- I don't care what it is. 508 00:23:49,140 --> 00:23:51,390 Let's just call it r equals g of theta. 509 00:23:51,390 --> 00:23:53,777 Now, the interesting point, again-- and I 510 00:23:53,777 --> 00:23:55,610 keep saying "interesting point" because they 511 00:23:55,610 --> 00:23:57,609 are interesting points-- that if you have really 512 00:23:57,609 --> 00:23:59,860 taken Part 1 of this course seriously, 513 00:23:59,860 --> 00:24:03,120 you're going to be amazed to see how much free mileage 514 00:24:03,120 --> 00:24:06,630 you get out of Part 2 just by translating things back 515 00:24:06,630 --> 00:24:09,580 in to our general theorems of Part 1. 516 00:24:09,580 --> 00:24:12,570 Let me see how I could find the area of this particular region. 517 00:24:12,570 --> 00:24:16,575 The idea is I'll take a little increment of area 518 00:24:16,575 --> 00:24:18,390 here between these two black lines. 519 00:24:21,100 --> 00:24:23,630 Remember, my basic building blocks are now r values. 520 00:24:23,630 --> 00:24:26,160 What I'll do now is I'll pick the smallest 521 00:24:26,160 --> 00:24:30,390 value of r that gives me an arc that lies inside this segment. 522 00:24:30,390 --> 00:24:32,880 then I'll pick the biggest value of r 523 00:24:32,880 --> 00:24:34,360 that encloses this segment. 524 00:24:34,360 --> 00:24:39,450 Notice, what I do is I'll let capital R sub capital 525 00:24:39,450 --> 00:24:40,590 M denote this radius. 526 00:24:40,590 --> 00:24:43,300 See, that's this distance from here to here. 527 00:24:43,300 --> 00:24:47,270 I'll let little r sub little m represent this distance. 528 00:24:47,270 --> 00:24:51,890 What I'm saying is, if I now swing two arcs, 529 00:24:51,890 --> 00:24:53,920 my area theorems from the first semester-- 530 00:24:53,920 --> 00:24:56,580 my area axioms from the first part of course-- 531 00:24:56,580 --> 00:24:57,300 are still valid. 532 00:24:57,300 --> 00:25:00,730 Namely, the smaller r-- the smaller sector-- 533 00:25:00,730 --> 00:25:04,090 is contained inside the delta A that I'm looking for. 534 00:25:04,090 --> 00:25:06,750 And the larger sector contains the region 535 00:25:06,750 --> 00:25:08,720 that I'm looking for. 536 00:25:08,720 --> 00:25:12,020 In other words, the region that I'm looking for, delta A, 537 00:25:12,020 --> 00:25:16,580 is caught between the areas of these two sectors. 538 00:25:16,580 --> 00:25:18,540 Now, how do you find the area of a sector? 539 00:25:18,540 --> 00:25:21,710 What you do is you take the area of the entire circle 540 00:25:21,710 --> 00:25:25,650 and divide it by the fractional part of the circle 541 00:25:25,650 --> 00:25:26,940 that you're taking. 542 00:25:26,940 --> 00:25:30,540 For example, the area of the circle whose radius is capital 543 00:25:30,540 --> 00:25:33,930 R sub M is pi R sub M squared. 544 00:25:33,930 --> 00:25:36,710 Now, what portion of the circle am I taking? 545 00:25:36,710 --> 00:25:38,410 The angle is delta theta. 546 00:25:38,410 --> 00:25:42,780 I'm using radian measure, so the entire swinging angle would 547 00:25:42,780 --> 00:25:46,490 have been 2*pi, so I'm taking delta theta over 2*pi 548 00:25:46,490 --> 00:25:47,860 of the entire circle. 549 00:25:47,860 --> 00:25:50,780 So that's the area of the larger sector. 550 00:25:50,780 --> 00:25:53,440 What is the area of the smaller sector? 551 00:25:53,440 --> 00:25:55,400 The area of the smaller sector is 552 00:25:55,400 --> 00:25:59,130 the area of the entire circle-- pi r sub little m squared-- 553 00:25:59,130 --> 00:26:00,460 times, again, what? 554 00:26:00,460 --> 00:26:02,740 Delta theta over 2*pi. 555 00:26:02,740 --> 00:26:07,800 And because the region is embedded between these two, 556 00:26:07,800 --> 00:26:10,640 its area is caught between these two areas. 557 00:26:10,640 --> 00:26:12,570 And if I now solve for the delta A, 558 00:26:12,570 --> 00:26:16,210 I simplify, I cancel the pi's, I get that delta A is 559 00:26:16,210 --> 00:26:17,170 caught between what? 560 00:26:17,170 --> 00:26:21,990 1/2 capital R sub M squared delta theta and 1/2 little r 561 00:26:21,990 --> 00:26:23,920 sub m squared delta theta. 562 00:26:23,920 --> 00:26:26,540 I now divide through by delta theta. 563 00:26:26,540 --> 00:26:29,060 And I make a non-crucial assumption here 564 00:26:29,060 --> 00:26:31,630 that delta theta is greater than 0, remembering 565 00:26:31,630 --> 00:26:33,790 that delta theta is negative. 566 00:26:33,790 --> 00:26:36,900 All I have to do is reverse the signs of the inequalities. 567 00:26:36,900 --> 00:26:39,250 The only reason I assumed that delta theta was positive 568 00:26:39,250 --> 00:26:42,100 is so that I wouldn't change the direction of the inequalities. 569 00:26:42,100 --> 00:26:44,470 A similar demonstration will hold 570 00:26:44,470 --> 00:26:46,090 when delta theta is negative. 571 00:26:46,090 --> 00:26:48,900 The important point is I then divide through by delta theta. 572 00:26:48,900 --> 00:26:50,980 I get delta A over delta theta. 573 00:26:50,980 --> 00:26:54,430 It's caught between 1/2 little r sub m squared and 1/2 574 00:26:54,430 --> 00:26:56,460 capitalized R sub M squared. 575 00:26:56,460 --> 00:26:59,361 Now, as I let delta theta approach 0, what does that 576 00:26:59,361 --> 00:26:59,860 mean? 577 00:26:59,860 --> 00:27:02,570 I'm going to let delta theta close in over here-- 578 00:27:02,570 --> 00:27:03,130 approach 0. 579 00:27:03,130 --> 00:27:04,830 What's happening here? 580 00:27:04,830 --> 00:27:06,530 This is a fixed value of r. 581 00:27:06,530 --> 00:27:10,770 Notice that little r sub m and capital R sub M, 582 00:27:10,770 --> 00:27:12,830 as delta theta approaches 0, they're 583 00:27:12,830 --> 00:27:15,440 both being pushed closer and closer to r. 584 00:27:15,440 --> 00:27:18,360 In other words, as delta theta approaches 0, 585 00:27:18,360 --> 00:27:22,520 r sub little m and capital R sub M 586 00:27:22,520 --> 00:27:29,952 both approach r, provided that r is a continuous function 587 00:27:29,952 --> 00:27:31,660 of theta-- in other words, that there are 588 00:27:31,660 --> 00:27:33,770 no breaks in the curve here. 589 00:27:33,770 --> 00:27:35,810 OK? 590 00:27:35,810 --> 00:27:37,500 The important point, then, is what? 591 00:27:37,500 --> 00:27:40,470 We can then take the limits as delta theta approaches 0. 592 00:27:40,470 --> 00:27:42,970 And we find that dA / d theta is the limit, 593 00:27:42,970 --> 00:27:45,600 as delta theta approaches 0, delta A divided 594 00:27:45,600 --> 00:27:48,170 by delta theta, which simply is what now? 595 00:27:48,170 --> 00:27:49,360 We come back here. 596 00:27:49,360 --> 00:27:52,830 As delta theta approaches 0, little r 597 00:27:52,830 --> 00:27:56,740 sub m and capital R sub M both approach r, 598 00:27:56,740 --> 00:27:58,800 and therefore, this common limit becomes 599 00:27:58,800 --> 00:28:02,870 1/2 r squared, therefore integrating 600 00:28:02,870 --> 00:28:05,330 this between what limits? 601 00:28:05,330 --> 00:28:09,630 Between theta_1 and theta_2, I find that the area of my region 602 00:28:09,630 --> 00:28:13,610 is the integral from theta_1 to theta_2, 1/2 r squared 603 00:28:13,610 --> 00:28:15,880 d theta, which, of course, means what? 604 00:28:15,880 --> 00:28:21,880 This integral 1/2-- r is g of theta, 605 00:28:21,880 --> 00:28:23,990 so I square that-- times d theta. 606 00:28:23,990 --> 00:28:27,030 And notice that this is a function of theta alone. 607 00:28:27,030 --> 00:28:29,500 And so I can find this particular area. 608 00:28:29,500 --> 00:28:32,904 What's crucial to understand is that the area does not 609 00:28:32,904 --> 00:28:34,320 change just because you're dealing 610 00:28:34,320 --> 00:28:37,210 with polar coordinates rather than Cartesian coordinates. 611 00:28:37,210 --> 00:28:39,860 But rather the form of the equation changes. 612 00:28:39,860 --> 00:28:41,980 What I mean by that is something like this. 613 00:28:41,980 --> 00:28:43,710 Let's suppose I have a region like this. 614 00:28:46,930 --> 00:28:48,077 This region is inanimate. 615 00:28:48,077 --> 00:28:50,660 It doesn't know whether you're looking at in polar coordinates 616 00:28:50,660 --> 00:28:52,470 or in Cartesian coordinates. 617 00:28:52,470 --> 00:28:55,950 In Cartesian coordinates, it's bounded above by the curve 618 00:28:55,950 --> 00:28:59,850 y equals f_1 of x, and it's bounded below by the curve 619 00:28:59,850 --> 00:29:02,139 y equals f_2 of x. 620 00:29:02,139 --> 00:29:03,930 From what we studied in the first semester, 621 00:29:03,930 --> 00:29:07,410 the area of the region r in Cartesian coordinates 622 00:29:07,410 --> 00:29:14,140 is the integral from 0 to A, f_1 of x minus f_2 of x, dx. 623 00:29:14,140 --> 00:29:17,960 On the other hand, if I call this curve r equals g of theta, 624 00:29:17,960 --> 00:29:21,050 in terms of polar coordinates, and the initial angle is 625 00:29:21,050 --> 00:29:24,520 theta_1, and the terminal angle here is theta_2, 626 00:29:24,520 --> 00:29:27,380 then the area of that same region r 627 00:29:27,380 --> 00:29:32,180 is given to be 1/2 integral from theta_1 to theta_2, g of theta 628 00:29:32,180 --> 00:29:34,610 squared d theta. 629 00:29:34,610 --> 00:29:37,610 Mathematically, this integral in terms 630 00:29:37,610 --> 00:29:40,850 of x and this integral in terms of theta 631 00:29:40,850 --> 00:29:43,340 look completely different. 632 00:29:43,340 --> 00:29:45,790 But the crucial point is they are simply 633 00:29:45,790 --> 00:29:49,310 different expressions for the same answer. 634 00:29:49,310 --> 00:29:51,700 Which of the two is the better one to use? 635 00:29:51,700 --> 00:29:54,030 It depends on the particular problem. 636 00:29:54,030 --> 00:29:56,980 If, for example, a problem begs for 637 00:29:56,980 --> 00:29:59,070 a polar coordinates interpretation, 638 00:29:59,070 --> 00:30:00,840 use polar coordinates. 639 00:30:00,840 --> 00:30:04,310 In fact, polar coordinates and Cartesian coordinates 640 00:30:04,310 --> 00:30:06,560 can both be bad sets of equations. 641 00:30:06,560 --> 00:30:07,570 Who knows? 642 00:30:07,570 --> 00:30:08,890 It's not the point. 643 00:30:08,890 --> 00:30:11,610 The point is that we now have two coordinate systems. 644 00:30:11,610 --> 00:30:14,450 There are others that we will define as the course goes on. 645 00:30:14,450 --> 00:30:16,710 But the important point is that we are now 646 00:30:16,710 --> 00:30:21,870 ready to tackle motion in the plane for central force fields 647 00:30:21,870 --> 00:30:23,720 if we so desire. 648 00:30:23,720 --> 00:30:26,410 At this particular moment, I do so desire. 649 00:30:26,410 --> 00:30:28,230 So the chances are that next time, 650 00:30:28,230 --> 00:30:32,310 we will be talking about velocity and acceleration 651 00:30:32,310 --> 00:30:34,730 vectors in polar coordinates. 652 00:30:34,730 --> 00:30:39,160 At any rate, until next time, good bye. 653 00:30:39,160 --> 00:30:41,540 Funding for the publication of this video 654 00:30:41,540 --> 00:30:46,410 was provided by the Gabriella and Paul Rosenbaum Foundation. 655 00:30:46,410 --> 00:30:50,590 Help OCW continue to provide free and open access to MIT 656 00:30:50,590 --> 00:30:58,290 courses by making a donation at ocw.mit.edu/donate.