1 00:00:00,060 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,950 Commons license. 3 00:00:03,950 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:10,230 continue to offer high quality educational resources for free. 5 00:00:10,230 --> 00:00:12,550 To make a donation or to view additional materials 6 00:00:12,550 --> 00:00:15,850 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,850 --> 00:00:21,950 at ocw.mit.edu. 8 00:00:21,950 --> 00:00:25,000 PROFESSOR: Today we're going to continue our discussion 9 00:00:25,000 --> 00:00:26,780 of parametric curves. 10 00:00:26,780 --> 00:00:29,960 I have to tell you about arc length. 11 00:00:29,960 --> 00:00:33,590 And let me remind me where we left off last time. 12 00:00:33,590 --> 00:00:45,820 This is parametric curves, continued. 13 00:00:45,820 --> 00:00:50,380 Last time, we talked about the parametric representation 14 00:00:50,380 --> 00:00:51,730 for the circle. 15 00:00:51,730 --> 00:00:55,960 Or one of the parametric representations for the circle. 16 00:00:55,960 --> 00:00:59,210 Which was this one here. 17 00:00:59,210 --> 00:01:05,590 And first we noted that this does parameterize, 18 00:01:05,590 --> 00:01:07,770 as we say, the circle. 19 00:01:07,770 --> 00:01:10,390 That satisfies the equation for the circle. 20 00:01:10,390 --> 00:01:17,990 And it's traced counterclockwise. 21 00:01:17,990 --> 00:01:20,590 The picture looks like this. 22 00:01:20,590 --> 00:01:22,350 Here's the circle. 23 00:01:22,350 --> 00:01:25,400 And it starts out here at t = 0 and it gets up 24 00:01:25,400 --> 00:01:31,820 to here at time t = pi / 2. 25 00:01:31,820 --> 00:01:41,330 So now I have to talk to you about arc length. 26 00:01:41,330 --> 00:01:43,840 In this parametric form. 27 00:01:43,840 --> 00:01:46,170 And the results should be the same as arc length 28 00:01:46,170 --> 00:01:48,830 around this circle ordinarily. 29 00:01:48,830 --> 00:01:55,840 And we start out with this basic differential relationship. 30 00:01:55,840 --> 00:02:00,080 ds^2 is dx^2 + dy^2. 31 00:02:00,080 --> 00:02:04,880 And then I'm going to take the square root, divide by dt, 32 00:02:04,880 --> 00:02:08,540 so the rate of change with respect to t of s 33 00:02:08,540 --> 00:02:10,780 is going to be the square root. 34 00:02:10,780 --> 00:02:13,220 Well, maybe I'll write it without dividing. 35 00:02:13,220 --> 00:02:15,130 Just write it as ds. 36 00:02:15,130 --> 00:02:24,640 So this would be (dx/dt)^2 + (dy/dt)^2, dt. 37 00:02:24,640 --> 00:02:27,050 So this is what you get formally from this equation. 38 00:02:27,050 --> 00:02:28,920 If you take its square roots and you 39 00:02:28,920 --> 00:02:32,470 divide by dt squared in the-- inside 40 00:02:32,470 --> 00:02:35,510 the square root, and you multiply by dt outside, 41 00:02:35,510 --> 00:02:36,630 so that those cancel. 42 00:02:36,630 --> 00:02:39,220 And this is the formal connection between the two. 43 00:02:39,220 --> 00:02:41,600 We'll be saying just a few more words 44 00:02:41,600 --> 00:02:48,430 in a few minutes about how to make sense of that rigorously. 45 00:02:48,430 --> 00:02:55,020 Alright so that's the set of formulas for the infinitesimal, 46 00:02:55,020 --> 00:02:57,110 the differential of arc length. 47 00:02:57,110 --> 00:03:00,820 And so to figure it out, I have to differentiate x with respect 48 00:03:00,820 --> 00:03:02,810 to t. 49 00:03:02,810 --> 00:03:04,225 And remember x is up here. 50 00:03:04,225 --> 00:03:11,330 It's defined by a cos t, so its derivative is -a sin t. 51 00:03:11,330 --> 00:03:19,850 And similarly, dy/dt = a cos t. 52 00:03:19,850 --> 00:03:22,000 And so I can plug this in. 53 00:03:22,000 --> 00:03:23,620 And I get the arc length element, 54 00:03:23,620 --> 00:03:36,920 which is the square root of (-a sin t)^2 + (a cos t)^2, dt. 55 00:03:36,920 --> 00:03:44,670 Which just becomes the square root of a^2, dt, or a dt. 56 00:03:44,670 --> 00:03:46,370 Now, I was about to divide by t. 57 00:03:46,370 --> 00:03:48,630 Let me do that now. 58 00:03:48,630 --> 00:03:52,050 We can also write the rate of change of arc length 59 00:03:52,050 --> 00:03:53,430 with respect to t. 60 00:03:53,430 --> 00:03:55,640 And that's a, in this case. 61 00:03:55,640 --> 00:04:01,420 And this gets interpreted as the speed 62 00:04:01,420 --> 00:04:03,180 of the particle going around. 63 00:04:03,180 --> 00:04:07,270 So not only, let me trade these two guys, 64 00:04:07,270 --> 00:04:14,020 not only do we have the direction is counterclockwise, 65 00:04:14,020 --> 00:04:20,010 but we also have that the speed is, if you like, it's uniform. 66 00:04:20,010 --> 00:04:21,740 It's constant speed. 67 00:04:21,740 --> 00:04:23,650 And the rate is a. 68 00:04:23,650 --> 00:04:26,810 So that's ds/dt. 69 00:04:26,810 --> 00:04:30,840 Travelling around. 70 00:04:30,840 --> 00:04:34,310 And that means that we can play around with the speed. 71 00:04:34,310 --> 00:04:37,650 And I just want to point out-- So the standard thing, what 72 00:04:37,650 --> 00:04:39,430 you'll have to get used to, and this 73 00:04:39,430 --> 00:04:42,520 is a standard presentation, you'll see this everywhere. 74 00:04:42,520 --> 00:04:45,710 In your physics classes and your other math classes, 75 00:04:45,710 --> 00:04:51,230 if you want to change the speed, so a new speed 76 00:04:51,230 --> 00:05:01,440 going around this would be, if I set up the equations this way. 77 00:05:01,440 --> 00:05:05,650 Now I'm tracing around the same circle. 78 00:05:05,650 --> 00:05:08,120 But the speed is going to turn out 79 00:05:08,120 --> 00:05:11,470 to be, if you figure it out, there'll 80 00:05:11,470 --> 00:05:13,200 be an extra factor of k. 81 00:05:13,200 --> 00:05:16,460 So it'll be ak. 82 00:05:16,460 --> 00:05:19,540 That's what we'll work out to be the speed. 83 00:05:19,540 --> 00:05:22,290 Provided k is positive and a is positive. 84 00:05:22,290 --> 00:05:30,490 So we're making these conventions. 85 00:05:30,490 --> 00:05:37,180 The constants that we're using are positive. 86 00:05:37,180 --> 00:05:40,980 Now, that's the first and most basic example. 87 00:05:40,980 --> 00:05:42,730 The one that comes up constantly. 88 00:05:42,730 --> 00:05:46,120 Now, let me just make those comments about notation 89 00:05:46,120 --> 00:05:47,860 that I wanted to make. 90 00:05:47,860 --> 00:05:51,884 And we've been treating these squared differentials here 91 00:05:51,884 --> 00:05:53,300 for a little while and I just want 92 00:05:53,300 --> 00:05:54,810 to pay attention a little bit more 93 00:05:54,810 --> 00:05:57,280 carefully to these manipulations. 94 00:05:57,280 --> 00:05:59,660 And what's allowed and what's not. 95 00:05:59,660 --> 00:06:01,890 And what's justified and what's not. 96 00:06:01,890 --> 00:06:06,680 So the basis for this was this approximate calculation that we 97 00:06:06,680 --> 00:06:11,440 had, that (delta s)^2 was (delta x)^2 + (delta y)^2. 98 00:06:11,440 --> 00:06:16,370 This is how we justified the arc length formula before. 99 00:06:16,370 --> 00:06:19,530 And let me just show you that the formula that I 100 00:06:19,530 --> 00:06:22,410 have up here, this basic formula for arc 101 00:06:22,410 --> 00:06:24,760 length in the parametric form, follows 102 00:06:24,760 --> 00:06:26,520 just as the other one did. 103 00:06:26,520 --> 00:06:31,370 And now I'm going to do it slightly more rigorously. 104 00:06:31,370 --> 00:06:34,400 I do the division really in disguise before I take 105 00:06:34,400 --> 00:06:36,310 the limit of the infinitesimal. 106 00:06:36,310 --> 00:06:40,349 So all I'm really doing is I'm doing this. 107 00:06:40,349 --> 00:06:42,390 Dividing through by this, and sorry this is still 108 00:06:42,390 --> 00:06:43,340 approximately equal. 109 00:06:43,340 --> 00:06:46,760 So I'm not dividing by something that's 0 or infinitesimal. 110 00:06:46,760 --> 00:06:49,320 I'm dividing by something nonzero. 111 00:06:49,320 --> 00:06:54,120 And here I have ((delta x)/(delta t))^2 + ((delta 112 00:06:54,120 --> 00:06:58,560 y)/(delta t))^2 And then in the limit, 113 00:06:58,560 --> 00:07:04,170 I have ds/dt is equal to the square root of this guy. 114 00:07:04,170 --> 00:07:13,820 Or, if you like, the square of it, so. 115 00:07:13,820 --> 00:07:17,200 So it's legal to divide by something that's almost 0 116 00:07:17,200 --> 00:07:20,340 and then take the limit as we go to 0. 117 00:07:20,340 --> 00:07:22,280 This is really what derivatives are all about. 118 00:07:22,280 --> 00:07:24,850 That we get a limit here. 119 00:07:24,850 --> 00:07:27,010 As the denominator goes to 0. 120 00:07:27,010 --> 00:07:31,510 Because the numerator's going to 0 too. 121 00:07:31,510 --> 00:07:32,770 So that's the notation. 122 00:07:32,770 --> 00:07:38,060 And now I want to warn you, maybe just a little bit, 123 00:07:38,060 --> 00:07:42,420 about misuses, if you like, of the notation. 124 00:07:42,420 --> 00:07:45,820 We don't do absolutely everything this way. 125 00:07:45,820 --> 00:07:49,280 This expression that came up with the squares, 126 00:07:49,280 --> 00:07:55,130 you should never write it as this. 127 00:07:55,130 --> 00:08:01,600 This, put it on the board but very quickly, never. 128 00:08:01,600 --> 00:08:02,420 OK. 129 00:08:02,420 --> 00:08:07,130 Don't do that. 130 00:08:07,130 --> 00:08:08,700 We use these square differentials, 131 00:08:08,700 --> 00:08:12,840 but we don't do it with these ratios here. 132 00:08:12,840 --> 00:08:15,900 But there was another place which is slightly confusing. 133 00:08:15,900 --> 00:08:17,680 It looks very similar, where we did 134 00:08:17,680 --> 00:08:20,247 use the square of the differential in a denominator. 135 00:08:20,247 --> 00:08:22,580 And I just want to point out to you that it's different. 136 00:08:22,580 --> 00:08:23,940 It's not the same. 137 00:08:23,940 --> 00:08:25,810 And it is OK. 138 00:08:25,810 --> 00:08:31,660 And that was this one. 139 00:08:31,660 --> 00:08:33,970 This thing here. 140 00:08:33,970 --> 00:08:36,900 This is a second derivative, it's something else. 141 00:08:36,900 --> 00:08:39,300 And it's got a dt^2 in the denominator. 142 00:08:39,300 --> 00:08:41,230 So it looks rather similar. 143 00:08:41,230 --> 00:08:49,230 But what this represents is the quantity d/dt squared. 144 00:08:49,230 --> 00:08:51,840 And you can see the squares came in. 145 00:08:51,840 --> 00:08:53,790 And squared the two expressions. 146 00:08:53,790 --> 00:08:58,970 And then there's also an x over here. 147 00:08:58,970 --> 00:09:00,600 So that's legal. 148 00:09:00,600 --> 00:09:02,550 Those are notations that we do use. 149 00:09:02,550 --> 00:09:04,070 And we can even calculate this. 150 00:09:04,070 --> 00:09:05,650 It has a perfectly good meaning. 151 00:09:05,650 --> 00:09:07,540 It's the same as the derivative with respect 152 00:09:07,540 --> 00:09:10,870 to t of the derivative of x, which we already know was minus 153 00:09:10,870 --> 00:09:17,530 sine-- sorry, a sin t, I guess. 154 00:09:17,530 --> 00:09:21,120 Not this example, but the previous one. 155 00:09:21,120 --> 00:09:21,870 Up here. 156 00:09:21,870 --> 00:09:24,340 So the derivative is this and so I can 157 00:09:24,340 --> 00:09:26,140 differentiate a second time. 158 00:09:26,140 --> 00:09:29,850 And I get -a cos t. 159 00:09:29,850 --> 00:09:31,760 So that's a perfectly legal operation. 160 00:09:31,760 --> 00:09:33,440 Everything in there makes sense. 161 00:09:33,440 --> 00:09:39,880 Just don't use that. 162 00:09:39,880 --> 00:09:41,770 There's another really unfortunate thing, 163 00:09:41,770 --> 00:09:45,820 right which is that the 2 creeps in funny places with sines. 164 00:09:45,820 --> 00:09:48,080 You have sine squared. 165 00:09:48,080 --> 00:09:50,150 It would be out here, it comes up here 166 00:09:50,150 --> 00:09:51,810 for some strange reason. 167 00:09:51,810 --> 00:09:54,730 This is just because typographers are lazy 168 00:09:54,730 --> 00:09:56,880 or somebody somewhere in the history 169 00:09:56,880 --> 00:10:00,620 of mathematical typography decided to let the 2 migrate. 170 00:10:00,620 --> 00:10:04,650 It would be like putting the 2 over here. 171 00:10:04,650 --> 00:10:07,940 There's inconsistency in mathematics, right. 172 00:10:07,940 --> 00:10:11,570 We're not perfect and people just develop these notations. 173 00:10:11,570 --> 00:10:14,450 So we have to live with them. 174 00:10:14,450 --> 00:10:20,210 The ones that people accept as conventions. 175 00:10:20,210 --> 00:10:23,230 The next example that I want to give you 176 00:10:23,230 --> 00:10:24,920 is just slightly different. 177 00:10:24,920 --> 00:10:29,140 It'll be a non-constant speed parameterization. 178 00:10:29,140 --> 00:10:32,470 Here x = 2 sin t. 179 00:10:32,470 --> 00:10:37,590 And y is, say, cos t. 180 00:10:37,590 --> 00:10:40,450 And let's keep track of what this one does. 181 00:10:40,450 --> 00:10:43,605 Now, this is a skill which I'm going 182 00:10:43,605 --> 00:10:45,170 to ask you about quite a bit. 183 00:10:45,170 --> 00:10:46,690 And it's one of several skills. 184 00:10:46,690 --> 00:10:48,970 You'll have to connect this with some kind 185 00:10:48,970 --> 00:10:50,330 of rectangular equation. 186 00:10:50,330 --> 00:10:51,725 An equation for x and y. 187 00:10:51,725 --> 00:10:54,230 And we'll be doing a certain amount of this today. 188 00:10:54,230 --> 00:10:56,240 In another context. 189 00:10:56,240 --> 00:11:00,510 Right here, to see the pattern, we know that the relationship 190 00:11:00,510 --> 00:11:04,160 we're going to want to use is that sin^2 + cos^2 = 1. 191 00:11:04,160 --> 00:11:11,910 So in fact the right thing to do here is to take 1/4 x^2 + y^2. 192 00:11:11,910 --> 00:11:17,020 And that's going to turn out to be sin^2 t + cos^2 t. 193 00:11:17,020 --> 00:11:18,310 Which is 1. 194 00:11:18,310 --> 00:11:19,500 So there's the equation. 195 00:11:19,500 --> 00:11:24,690 Here's the rectangular equation for this parametric curve. 196 00:11:24,690 --> 00:11:32,030 And this describes an ellipse. 197 00:11:32,030 --> 00:11:35,570 That's not the only information that we can get here. 198 00:11:35,570 --> 00:11:37,180 The other information that we can get 199 00:11:37,180 --> 00:11:39,570 is this qualitative information of where 200 00:11:39,570 --> 00:11:42,360 we start, where we're going, the direction. 201 00:11:42,360 --> 00:11:46,540 It starts out, I claim, at t = 0. 202 00:11:46,540 --> 00:11:54,630 That's when t = 0, this is (2 sin 0, cos 0), right? 203 00:11:54,630 --> 00:12:00,330 (2 sin 0, cos 0) is equal to the point (0, 1). 204 00:12:00,330 --> 00:12:02,360 So it starts up up here. 205 00:12:02,360 --> 00:12:05,140 At (0, 1). 206 00:12:05,140 --> 00:12:08,520 And then the next little place, so this is one thing 207 00:12:08,520 --> 00:12:11,400 that certainly you want to do. t = pi/2 208 00:12:11,400 --> 00:12:14,510 is maybe the next easy point to plot. 209 00:12:14,510 --> 00:12:22,830 And that's going to be (2 sin(pi/2), cos(pi/2)). 210 00:12:22,830 --> 00:12:27,880 And that's just (2, 0). 211 00:12:27,880 --> 00:12:31,490 And so that's over here somewhere. 212 00:12:31,490 --> 00:12:34,422 This is (2, 0). 213 00:12:34,422 --> 00:12:36,130 And we know it travels along the ellipse. 214 00:12:36,130 --> 00:12:40,120 And we know the minor axis is 1, and the major axis is 2, 215 00:12:40,120 --> 00:12:43,000 so it's doing this. 216 00:12:43,000 --> 00:12:45,090 So this is what happens at t = 0. 217 00:12:45,090 --> 00:12:48,390 This is where we are at t = pi/2. 218 00:12:48,390 --> 00:12:51,510 And it continues all the way around, etc. 219 00:12:51,510 --> 00:12:53,370 To the rest of the ellipse. 220 00:12:53,370 --> 00:12:57,750 This is the direction. 221 00:12:57,750 --> 00:13:09,620 So this one happens to be clockwise. 222 00:13:09,620 --> 00:13:12,640 Alright, now let's keep track of its speed. 223 00:13:12,640 --> 00:13:25,610 Let's keep track of the speed, and also the arc length. 224 00:13:25,610 --> 00:13:32,830 So the speed is the square root of the derivatives here. 225 00:13:32,830 --> 00:13:38,580 That would be (2 cos t)^2 + (sin t)^2. 226 00:13:42,160 --> 00:13:48,060 And the arc length is what? 227 00:13:48,060 --> 00:13:49,840 Well, if we want to go all the way around, 228 00:13:49,840 --> 00:13:53,540 we need to know that that takes a total of 2 pi. 229 00:13:53,540 --> 00:13:55,990 So 0 to 2 pi. 230 00:13:55,990 --> 00:13:59,310 And then we have to integrate ds, which is this expression, 231 00:13:59,310 --> 00:14:02,620 or ds/dt, dt. 232 00:14:02,620 --> 00:14:11,630 So that's the square root of 4 cos^2 t + sin^2 t, dt. 233 00:14:20,820 --> 00:14:26,580 The bad news, if you like, is that this is not 234 00:14:26,580 --> 00:14:38,524 an elementary integral. 235 00:14:38,524 --> 00:14:39,940 In other words, no matter how long 236 00:14:39,940 --> 00:14:44,240 you try to figure out how to antidifferentiate 237 00:14:44,240 --> 00:14:47,180 this expression, no matter how many substitutions you try, 238 00:14:47,180 --> 00:14:50,420 you will fail. 239 00:14:50,420 --> 00:14:52,030 That's the bad news. 240 00:14:52,030 --> 00:14:58,430 The good news is this is not an elementary integral. 241 00:14:58,430 --> 00:14:59,810 It's not an elementary integral. 242 00:14:59,810 --> 00:15:03,330 Which means that this is the answer to a question. 243 00:15:03,330 --> 00:15:06,230 Not something that you have to work on. 244 00:15:06,230 --> 00:15:11,680 So if somebody asks you for this arc length, you stop here. 245 00:15:11,680 --> 00:15:14,550 That's the answer, so it's actually better than it looks. 246 00:15:14,550 --> 00:15:17,700 And we'll try to-- I mean, I don't 247 00:15:17,700 --> 00:15:21,230 expect you to know already what all of the integrals 248 00:15:21,230 --> 00:15:22,680 are that are impossible. 249 00:15:22,680 --> 00:15:24,680 And which ones are hard and which ones are easy. 250 00:15:24,680 --> 00:15:27,200 So we'll try to coach you through when 251 00:15:27,200 --> 00:15:28,390 you face these things. 252 00:15:28,390 --> 00:15:31,670 It's not so easy to decide. 253 00:15:31,670 --> 00:15:34,345 I'll give you a few clues, but. 254 00:15:34,345 --> 00:15:34,845 OK. 255 00:15:34,845 --> 00:15:38,310 So this is the arc length. 256 00:15:38,310 --> 00:15:42,270 Now, I want to move on to the last thing that we did. 257 00:15:42,270 --> 00:15:44,410 Last type of thing that we did last time. 258 00:15:44,410 --> 00:15:54,140 Which is the surface area. 259 00:15:54,140 --> 00:15:55,230 And yeah, question. 260 00:15:55,230 --> 00:16:03,609 STUDENT: [INAUDIBLE] 261 00:16:03,609 --> 00:16:05,650 PROFESSOR: The question, this is a good question. 262 00:16:05,650 --> 00:16:08,290 The question is, when you draw the ellipse, 263 00:16:08,290 --> 00:16:11,650 do you not take into account what t is. 264 00:16:11,650 --> 00:16:16,460 The answer is that this is in disguise. 265 00:16:16,460 --> 00:16:20,360 What's going on here is we have a trouble 266 00:16:20,360 --> 00:16:24,720 with plotting in the plane what's really happening. 267 00:16:24,720 --> 00:16:29,210 So in other words, it's kind of in trouble. 268 00:16:29,210 --> 00:16:33,940 So the point is that we have two functions of t, not one. 269 00:16:33,940 --> 00:16:35,660 x(t) and y(t). 270 00:16:35,660 --> 00:16:38,970 So one thing that I can do if I plot things in the plane. 271 00:16:38,970 --> 00:16:41,710 In other words, the main point to make here 272 00:16:41,710 --> 00:16:45,150 is that we're not talking about the situation 273 00:16:45,150 --> 00:16:46,480 y is a function of x. 274 00:16:46,480 --> 00:16:47,870 We're out of that realm now. 275 00:16:47,870 --> 00:16:49,620 We're somewhere in a different part 276 00:16:49,620 --> 00:16:51,450 of the universe in our thought. 277 00:16:51,450 --> 00:16:54,760 And you should drop this point of view. 278 00:16:54,760 --> 00:16:56,990 So this depiction is not y as a function of x. 279 00:16:56,990 --> 00:17:00,940 Well, that's obvious because there are two values here, 280 00:17:00,940 --> 00:17:01,710 as opposed to one. 281 00:17:01,710 --> 00:17:02,960 So we're in trouble with that. 282 00:17:02,960 --> 00:17:04,750 And we have that background parameter, 283 00:17:04,750 --> 00:17:07,170 and that's exactly why we're using it. 284 00:17:07,170 --> 00:17:08,230 This parameter t. 285 00:17:08,230 --> 00:17:10,430 So that we can depict the entire curve. 286 00:17:10,430 --> 00:17:14,200 And deal with it as one thing. 287 00:17:14,200 --> 00:17:17,590 So since I can't really draw it, and since t is nowhere 288 00:17:17,590 --> 00:17:19,840 on the map, you should sort of imagine it as time, 289 00:17:19,840 --> 00:17:21,760 and there's some kind of trajectory which is travelling 290 00:17:21,760 --> 00:17:22,260 around. 291 00:17:22,260 --> 00:17:25,590 And then I just labelled a couple of the places. 292 00:17:25,590 --> 00:17:28,370 If somebody asked you to draw a picture of this, 293 00:17:28,370 --> 00:17:31,230 well, I'll tell you exactly where you need the picture 294 00:17:31,230 --> 00:17:33,250 in just one second, alright. 295 00:17:33,250 --> 00:17:36,690 It's going to come up right now in surface area. 296 00:17:36,690 --> 00:17:39,450 But otherwise, if nobody asks you to, 297 00:17:39,450 --> 00:17:44,020 you don't even have to put down t = 0 and t = pi / 2 here. 298 00:17:44,020 --> 00:17:46,402 Because nobody demanded it of you. 299 00:17:46,402 --> 00:17:47,110 Another question. 300 00:17:47,110 --> 00:17:51,842 STUDENT: [INAUDIBLE] 301 00:17:51,842 --> 00:17:53,550 PROFESSOR: So, another very good question 302 00:17:53,550 --> 00:17:55,890 which is exactly connected to this picture. 303 00:17:55,890 --> 00:17:58,060 So how is it that we're going to use the picture, 304 00:17:58,060 --> 00:18:02,190 and how is it we're going to use the notion of the t. 305 00:18:02,190 --> 00:18:07,040 The question was, why is this from t = 0 to t = 2 pi? 306 00:18:07,040 --> 00:18:11,000 That does use the t information on this diagram. 307 00:18:11,000 --> 00:18:13,150 the point is, we do know that t starts here. 308 00:18:13,150 --> 00:18:17,620 This is pi / 2, this is pi, this is 3 pi / 2, and this is 2 pi. 309 00:18:17,620 --> 00:18:19,380 When you go all the way around once, 310 00:18:19,380 --> 00:18:21,240 it's going to come back to itself. 311 00:18:21,240 --> 00:18:23,720 These are periodic functions of period 2 pi. 312 00:18:23,720 --> 00:18:26,780 And they come back to themselves exactly at 2 pi. 313 00:18:26,780 --> 00:18:29,030 And so that's why we know in order to get around once, 314 00:18:29,030 --> 00:18:32,250 we need to go from 0 to 2 pi. 315 00:18:32,250 --> 00:18:34,640 And the same thing is going to come up with surface area 316 00:18:34,640 --> 00:18:35,370 right now. 317 00:18:35,370 --> 00:18:38,720 That's going to be the issue, is what range of t 318 00:18:38,720 --> 00:18:45,620 we're going to need when we compute the surface area. 319 00:18:45,620 --> 00:18:52,144 STUDENT: [INAUDIBLE] 320 00:18:52,144 --> 00:18:54,185 PROFESSOR: In a question, what you might be asked 321 00:18:54,185 --> 00:18:56,180 is what's the rectangular equation 322 00:18:56,180 --> 00:18:57,650 for a parametric curve? 323 00:18:57,650 --> 00:19:01,720 So that would be 1/4 x^2 + y^2 = 1. 324 00:19:01,720 --> 00:19:03,430 And then you might be asked, plot it. 325 00:19:03,430 --> 00:19:06,960 Well, that would be a picture of the ellipse. 326 00:19:06,960 --> 00:19:10,380 OK, those are types of questions that are legal questions. 327 00:19:10,380 --> 00:19:27,439 STUDENT: [INAUDIBLE] 328 00:19:27,439 --> 00:19:28,980 PROFESSOR: The question is, do I need 329 00:19:28,980 --> 00:19:30,600 to know any specific formulas? 330 00:19:30,600 --> 00:19:33,250 Any formulas that you know and remember will help you. 331 00:19:33,250 --> 00:19:35,400 They may be of limited use. 332 00:19:35,400 --> 00:19:37,640 I'm not going to ask you to memorize anything 333 00:19:37,640 --> 00:19:40,820 except, I guarantee you that the circle is going to come up. 334 00:19:40,820 --> 00:19:43,500 Not the ellipse, the circle will come up everywhere 335 00:19:43,500 --> 00:19:44,320 in your life. 336 00:19:44,320 --> 00:19:47,710 So at least at MIT, your life at MIT. 337 00:19:47,710 --> 00:19:52,002 We're very round here. 338 00:19:52,002 --> 00:19:52,960 Yeah, another question. 339 00:19:52,960 --> 00:19:56,811 STUDENT: I'm just a tiny bit confused back to the basics. 340 00:19:56,811 --> 00:19:58,810 This is more a question from yesterday, I guess. 341 00:19:58,810 --> 00:20:04,390 But when you have your original ds^2 = dx^2 + dy^2, 342 00:20:04,390 --> 00:20:10,060 and then you integrate that to get arc length, how are you, 343 00:20:10,060 --> 00:20:14,360 the integral has dx's and dy's. 344 00:20:14,360 --> 00:20:18,590 So how are you just integrating with respect to dx? 345 00:20:18,590 --> 00:20:22,560 PROFESSOR: OK, the question is how are we just integrating 346 00:20:22,560 --> 00:20:24,200 with respect to x? 347 00:20:24,200 --> 00:20:26,880 So this is a question which goes back to last time. 348 00:20:26,880 --> 00:20:29,120 And what is it with arc length. 349 00:20:29,120 --> 00:20:30,370 So. 350 00:20:30,370 --> 00:20:35,442 I'm going to have to answer that question in connection 351 00:20:35,442 --> 00:20:36,400 with what we did today. 352 00:20:36,400 --> 00:20:38,420 So this is a subtle question. 353 00:20:38,420 --> 00:20:42,450 But I want you to realize that this is actually 354 00:20:42,450 --> 00:20:44,290 an important conceptual step here. 355 00:20:44,290 --> 00:20:49,810 So shhh, everybody, listen. 356 00:20:49,810 --> 00:20:53,340 If you're representing one-dimensional objects, 357 00:20:53,340 --> 00:20:56,050 which are curves, maybe, in space. 358 00:20:56,050 --> 00:20:58,270 Or in two dimensions. 359 00:20:58,270 --> 00:21:00,759 When you're keeping track of arc length, 360 00:21:00,759 --> 00:21:02,800 you're going to have to have an integral which is 361 00:21:02,800 --> 00:21:05,180 with respect to some variable. 362 00:21:05,180 --> 00:21:08,510 But that variable, you get to pick. 363 00:21:08,510 --> 00:21:12,310 And we're launching now into this variety of choices 364 00:21:12,310 --> 00:21:13,950 of variables with respect to which you 365 00:21:13,950 --> 00:21:15,980 can represent something. 366 00:21:15,980 --> 00:21:17,580 Now, there are some disadvantages 367 00:21:17,580 --> 00:21:19,370 on the circle to representing things 368 00:21:19,370 --> 00:21:21,480 with respect to the variable x. 369 00:21:21,480 --> 00:21:24,732 Because there are two points on the circle here. 370 00:21:24,732 --> 00:21:26,190 On the other hand, you actually can 371 00:21:26,190 --> 00:21:27,480 succeed with half the circle. 372 00:21:27,480 --> 00:21:29,620 So you can figure out the arc length that way. 373 00:21:29,620 --> 00:21:32,521 And then you can set it up as an integral dx. 374 00:21:32,521 --> 00:21:34,770 But you can also set it up as an integral with respect 375 00:21:34,770 --> 00:21:37,240 to any parameter you want. 376 00:21:37,240 --> 00:21:40,170 And the uniform parameter is perhaps the easiest one. 377 00:21:40,170 --> 00:21:43,210 This one is perhaps the easiest one. 378 00:21:43,210 --> 00:21:47,970 And so now the thing that's strange about this perspective 379 00:21:47,970 --> 00:21:51,470 - and I'm going to make this point later in the lecture 380 00:21:51,470 --> 00:21:55,810 as well - is that the letters x and y-- As I say, 381 00:21:55,810 --> 00:22:00,630 you should drop this notion that y is a function of x. 382 00:22:00,630 --> 00:22:03,950 This is what we're throwing away at this point. 383 00:22:03,950 --> 00:22:05,770 What we're thinking of is, you can 384 00:22:05,770 --> 00:22:08,110 describe things in terms of any coordinate you want. 385 00:22:08,110 --> 00:22:11,340 You just have to say what each one is in terms of the others. 386 00:22:11,340 --> 00:22:15,300 And these x and y over here are where 387 00:22:15,300 --> 00:22:18,380 we are in the Cartesian coordinate system. 388 00:22:18,380 --> 00:22:20,490 They're not-- And in this case they're 389 00:22:20,490 --> 00:22:24,610 functions of some other variable. 390 00:22:24,610 --> 00:22:25,720 Some other variable. 391 00:22:25,720 --> 00:22:27,150 So they're each functions. 392 00:22:27,150 --> 00:22:29,480 So the letters x and y just changed on you. 393 00:22:29,480 --> 00:22:33,710 They mean something different. x is no longer the variable. 394 00:22:33,710 --> 00:22:36,810 It's the function. 395 00:22:36,810 --> 00:22:38,542 Right? 396 00:22:38,542 --> 00:22:40,250 You're going to have to get used to that. 397 00:22:40,250 --> 00:22:42,380 That's because we run out of letters. 398 00:22:42,380 --> 00:22:44,870 And we kind of want to use all of them the way we want. 399 00:22:44,870 --> 00:22:48,290 I'll say some more about that later. 400 00:22:48,290 --> 00:22:51,220 So now I want to do this surface area example. 401 00:22:51,220 --> 00:22:59,150 I'm going to just take the surface area of the ellipsoid. 402 00:22:59,150 --> 00:23:11,340 The surface of the ellipsoid formed 403 00:23:11,340 --> 00:23:19,910 by revolving this previous example, which was Example 2. 404 00:23:19,910 --> 00:23:28,020 Around the y-axis. 405 00:23:28,020 --> 00:23:30,410 So we want to set up that surface area integral here 406 00:23:30,410 --> 00:23:32,490 for you. 407 00:23:32,490 --> 00:23:38,160 Now, I remind you that the area element looks like this. 408 00:23:38,160 --> 00:23:41,190 If you're revolving around the y-axis, 409 00:23:41,190 --> 00:23:42,815 that means you're going around this way 410 00:23:42,815 --> 00:23:43,720 and you have some curve. 411 00:23:43,720 --> 00:23:44,990 In this case it's this piece of an ellipse. 412 00:23:44,990 --> 00:23:46,475 If you sweep it around you're going 413 00:23:46,475 --> 00:23:48,770 to get what's called an ellipsoid. 414 00:23:48,770 --> 00:23:53,890 And there's a little chunk here, that you're wrapping around. 415 00:23:53,890 --> 00:23:58,430 And the important thing you need besides this ds, this arc 416 00:23:58,430 --> 00:24:04,120 length piece over here, is the distance to the axis. 417 00:24:04,120 --> 00:24:06,320 So that's this horizontal distance here. 418 00:24:06,320 --> 00:24:09,850 I'll draw it in another color. 419 00:24:09,850 --> 00:24:15,520 And that horizontal distance now has a name. 420 00:24:15,520 --> 00:24:18,670 And this is, again, the virtue of this coordinate system. 421 00:24:18,670 --> 00:24:20,170 The t is something else. 422 00:24:20,170 --> 00:24:21,020 This has a name. 423 00:24:21,020 --> 00:24:22,760 This distance has a name. 424 00:24:22,760 --> 00:24:27,080 This distance is called x. 425 00:24:27,080 --> 00:24:29,570 And it even has a formula. 426 00:24:29,570 --> 00:24:36,090 Its formula is 2 sin t. 427 00:24:36,090 --> 00:24:38,550 In terms of t. 428 00:24:38,550 --> 00:24:41,530 So the full formula up for the integral 429 00:24:41,530 --> 00:24:46,039 here is, I have to take the circumference when 430 00:24:46,039 --> 00:24:47,080 I spin this thing around. 431 00:24:47,080 --> 00:24:48,950 And this little arc length element. 432 00:24:48,950 --> 00:24:53,660 So I have here 2 pi times 2 sin t. 433 00:24:53,660 --> 00:24:55,640 That's the x variable here. 434 00:24:55,640 --> 00:25:00,560 And then I have here ds, which is kind of a mess. 435 00:25:00,560 --> 00:25:04,170 So unfortunately I don't quite have room for it. 436 00:25:04,170 --> 00:25:05,650 Plan ahead. 437 00:25:05,650 --> 00:25:15,200 Square root of 4 cos^2 t + sin^2 t, is that what it was, dt. 438 00:25:15,200 --> 00:25:17,740 Alright, I guess I squeezed it in there. 439 00:25:17,740 --> 00:25:20,090 So that was the arc length, which I re-copied 440 00:25:20,090 --> 00:25:21,620 from this board above. 441 00:25:21,620 --> 00:25:24,310 That was the ds piece. 442 00:25:24,310 --> 00:25:29,760 It's this whole thing including the dt. 443 00:25:29,760 --> 00:25:32,360 That's the answer except for one thing. 444 00:25:32,360 --> 00:25:33,590 What else do we need? 445 00:25:33,590 --> 00:25:35,350 We don't just need the integrand, 446 00:25:35,350 --> 00:25:37,720 this is half of setting up an integral. 447 00:25:37,720 --> 00:25:40,990 The other half of setting up an integral is the limits. 448 00:25:40,990 --> 00:25:42,840 We need specific limits here. 449 00:25:42,840 --> 00:25:46,760 Otherwise we don't have a number that we can get out. 450 00:25:46,760 --> 00:25:50,370 So we now have to think about what the limits are. 451 00:25:50,370 --> 00:25:52,550 And maybe somebody can see. 452 00:25:52,550 --> 00:25:54,429 It has something to do with this diagram 453 00:25:54,429 --> 00:25:55,470 of the ellipse over here. 454 00:25:55,470 --> 00:25:58,520 Can somebody guess what it is? 455 00:25:58,520 --> 00:25:59,480 0 to pi. 456 00:25:59,480 --> 00:26:02,070 Well, that was quick. 457 00:26:02,070 --> 00:26:02,620 That's it. 458 00:26:02,620 --> 00:26:04,709 Because we go from the top to the bottom, 459 00:26:04,709 --> 00:26:06,250 but we don't want to continue around. 460 00:26:06,250 --> 00:26:07,640 We don't want to go from 0 to 2 pi, 461 00:26:07,640 --> 00:26:09,723 because that would be duplicating what we're going 462 00:26:09,723 --> 00:26:12,020 to get when we spin around. 463 00:26:12,020 --> 00:26:13,730 And we know that we start at 0. 464 00:26:13,730 --> 00:26:15,540 It's interesting because it descends 465 00:26:15,540 --> 00:26:17,090 when you change variables to think 466 00:26:17,090 --> 00:26:20,360 of it in terms of the y variable it's going the opposite way. 467 00:26:20,360 --> 00:26:24,550 But anyway, just one piece of this is what we want. 468 00:26:24,550 --> 00:26:27,660 So that's this setup. 469 00:26:27,660 --> 00:26:36,230 And now I claim that this is actually a doable integral. 470 00:26:36,230 --> 00:26:37,850 However, it's long. 471 00:26:37,850 --> 00:26:39,830 I'm going to spare you, I'll just tell you 472 00:26:39,830 --> 00:26:41,330 how you would get started. 473 00:26:41,330 --> 00:26:45,970 You would use the substitution u = cos t. 474 00:26:45,970 --> 00:26:53,620 And then the du is going to be -sin t dt. 475 00:26:53,620 --> 00:26:56,290 But then, unfortunately, there's a lot more. 476 00:26:56,290 --> 00:26:57,810 There's another trig substitution 477 00:26:57,810 --> 00:27:01,430 with some other multiple of the cosine and so forth. 478 00:27:01,430 --> 00:27:02,420 So it goes on and on. 479 00:27:02,420 --> 00:27:06,260 If you want to check it yourself, you can. 480 00:27:06,260 --> 00:27:08,710 There's an inverse trig substitution which 481 00:27:08,710 --> 00:27:11,590 isn't compatible with this one. 482 00:27:11,590 --> 00:27:17,090 But it can be done. 483 00:27:17,090 --> 00:27:22,690 Calculated. 484 00:27:22,690 --> 00:27:26,980 In elementary terms. 485 00:27:26,980 --> 00:27:30,547 Yeah, another question. 486 00:27:30,547 --> 00:27:31,380 STUDENT: [INAUDIBLE] 487 00:27:31,380 --> 00:27:33,490 PROFESSOR: So, if you get this on an exam, 488 00:27:33,490 --> 00:27:35,240 I'm going to have to coach you through it. 489 00:27:35,240 --> 00:27:37,640 Either I'm going to have to tell you don't evaluate it 490 00:27:37,640 --> 00:27:40,130 or, you're going to have to work really hard. 491 00:27:40,130 --> 00:27:42,500 Or here's the first step, and then the next step 492 00:27:42,500 --> 00:27:44,299 is, keep on going. 493 00:27:44,299 --> 00:27:44,840 Or something. 494 00:27:44,840 --> 00:27:47,890 I'll have to give you some cues. 495 00:27:47,890 --> 00:27:49,260 Because it's quite long. 496 00:27:49,260 --> 00:27:52,860 This is way too long for an exam, this particular one. 497 00:27:52,860 --> 00:27:53,650 OK. 498 00:27:53,650 --> 00:27:55,359 It's not too long for a problem set. 499 00:27:55,359 --> 00:27:57,650 This is where I would leave you off if I were giving it 500 00:27:57,650 --> 00:27:58,320 to you on a problem set. 501 00:27:58,320 --> 00:28:00,220 Just to give you an idea of the order of magnitude. 502 00:28:00,220 --> 00:28:02,761 Whereas one of the ones that I did yesterday, I wouldn't even 503 00:28:02,761 --> 00:28:11,120 give you on a problem set, it was so long. 504 00:28:11,120 --> 00:28:17,630 So now, our next job is to move on to polar coordinates. 505 00:28:17,630 --> 00:28:20,960 Now, polar coordinates involve the geometry of circles. 506 00:28:20,960 --> 00:28:23,392 As I said, we really love circles here. 507 00:28:23,392 --> 00:28:24,100 We're very round. 508 00:28:24,100 --> 00:28:28,210 Just as I love 0, the rest of the Institute loves circles. 509 00:28:28,210 --> 00:28:47,380 So we're going to do that right now. 510 00:28:47,380 --> 00:28:58,900 What we're going to talk about now is polar coordinates. 511 00:28:58,900 --> 00:29:01,010 Which are set up in the following way. 512 00:29:01,010 --> 00:29:04,640 It's a way of describing the points in the plane. 513 00:29:04,640 --> 00:29:07,460 Here is a point in a plane, and here's 514 00:29:07,460 --> 00:29:10,530 what we think of as the usual x-y axes. 515 00:29:10,530 --> 00:29:12,860 And now this point is going to be described 516 00:29:12,860 --> 00:29:15,260 by a different pair of coordinates, different pair 517 00:29:15,260 --> 00:29:16,190 of numbers. 518 00:29:16,190 --> 00:29:26,420 Namely, the distance to the origin. 519 00:29:26,420 --> 00:29:30,490 And the second parameter here, second number here, 520 00:29:30,490 --> 00:29:32,550 is this angle theta. 521 00:29:32,550 --> 00:29:41,500 Which is the angle of ray from origin 522 00:29:41,500 --> 00:29:48,670 with the horizontal axis. 523 00:29:48,670 --> 00:29:50,620 So that's what it is in language. 524 00:29:50,620 --> 00:29:53,690 And you should put this in quotation marks, 525 00:29:53,690 --> 00:29:57,320 because it's not a perfect match. 526 00:29:57,320 --> 00:30:00,800 This is geometrically what you should always think of, 527 00:30:00,800 --> 00:30:03,720 but the technical details involve 528 00:30:03,720 --> 00:30:06,530 dealing directly with formulas. 529 00:30:06,530 --> 00:30:09,880 The first formula is the formula for x. 530 00:30:09,880 --> 00:30:11,590 And this is the fundamental, these two 531 00:30:11,590 --> 00:30:12,750 are the fundamental ones. 532 00:30:12,750 --> 00:30:16,120 Namely, x = r cos theta. 533 00:30:16,120 --> 00:30:17,860 The second formula is the formula 534 00:30:17,860 --> 00:30:21,380 for y, which is r sin theta. 535 00:30:21,380 --> 00:30:25,420 So these are the unambiguous definitions 536 00:30:25,420 --> 00:30:27,100 of polar coordinates. 537 00:30:27,100 --> 00:30:28,790 This is it. 538 00:30:28,790 --> 00:30:32,590 And this is the thing from which all other almost correct 539 00:30:32,590 --> 00:30:37,180 statements almost follow. 540 00:30:37,180 --> 00:30:39,320 But this is the one you should trust always. 541 00:30:39,320 --> 00:30:44,980 This is the unambiguous statement. 542 00:30:44,980 --> 00:30:47,360 So let me give you an example something that's 543 00:30:47,360 --> 00:30:52,040 close to being a good formula and is certainly 544 00:30:52,040 --> 00:30:57,530 useful in its way. 545 00:30:57,530 --> 00:31:04,180 Namely, you can think of r as being the square root of x^2 + 546 00:31:04,180 --> 00:31:05,810 y^2. 547 00:31:05,810 --> 00:31:07,336 That's easy enough to derive, it's 548 00:31:07,336 --> 00:31:08,460 the distance to the origin. 549 00:31:08,460 --> 00:31:11,320 That's pretty obvious. 550 00:31:11,320 --> 00:31:14,690 And the formula for theta, which you can also derive, 551 00:31:14,690 --> 00:31:17,480 which is that it's the inverse tangent of y y/x. 552 00:31:21,050 --> 00:31:24,310 However, let me just warn you that these formulas are 553 00:31:24,310 --> 00:31:26,870 slightly ambiguous. 554 00:31:26,870 --> 00:31:33,357 So somewhat ambiguous. 555 00:31:33,357 --> 00:31:35,440 In other words, you can't just apply them blindly. 556 00:31:35,440 --> 00:31:37,023 You actually have to look at a picture 557 00:31:37,023 --> 00:31:38,180 in order to get them right. 558 00:31:38,180 --> 00:31:43,690 In particular, r could be plus or minus here. 559 00:31:43,690 --> 00:31:47,950 And when you take the inverse tangent, 560 00:31:47,950 --> 00:31:52,510 there's an ambiguity between, it's the same as the inverse 561 00:31:52,510 --> 00:31:56,330 tangent of (-y)/(-x). 562 00:31:56,330 --> 00:32:00,550 So these minus signs are a plague on your existence. 563 00:32:00,550 --> 00:32:05,050 And you're not going to get a completely unambiguous answer 564 00:32:05,050 --> 00:32:07,760 out of these formulas without paying attention 565 00:32:07,760 --> 00:32:08,430 to the diagram. 566 00:32:08,430 --> 00:32:10,550 On the other hand, the formula up in the box 567 00:32:10,550 --> 00:32:14,337 there always works. 568 00:32:14,337 --> 00:32:15,920 So when people mean polar coordinates, 569 00:32:15,920 --> 00:32:17,370 they always mean that. 570 00:32:17,370 --> 00:32:22,370 And then they have conventions, which sometimes match things up 571 00:32:22,370 --> 00:32:27,550 with the formulas over on this next board. 572 00:32:27,550 --> 00:32:32,670 Let me give you various examples here first. 573 00:32:32,670 --> 00:32:36,260 But maybe first I should I should draw 574 00:32:36,260 --> 00:32:38,100 the two coordinate systems. 575 00:32:38,100 --> 00:32:40,560 So the coordinate system that we're used to 576 00:32:40,560 --> 00:32:43,360 is the rectangular coordinate system. 577 00:32:43,360 --> 00:32:49,190 And maybe I'll draw it in orange and green here. 578 00:32:49,190 --> 00:32:59,430 So these are the coordinate lines y = 0, y = 1, y = 2. 579 00:32:59,430 --> 00:33:01,950 That's how the coordinate system works. 580 00:33:01,950 --> 00:33:08,427 And over here we have the rest of the coordinate system. 581 00:33:08,427 --> 00:33:10,510 And this is the way we're thinking of x and y now. 582 00:33:10,510 --> 00:33:12,570 We're no longer thinking of y as a function of x and x 583 00:33:12,570 --> 00:33:13,986 as a function of y, we're thinking 584 00:33:13,986 --> 00:33:16,960 of x as a label of a place in a plane. 585 00:33:16,960 --> 00:33:20,900 And y as a label of a place in a plane. 586 00:33:20,900 --> 00:33:27,770 So here we have x = 0, x = 1, x = 2, etc. 587 00:33:27,770 --> 00:33:30,740 Here's x = -1. 588 00:33:30,740 --> 00:33:31,900 So forth. 589 00:33:31,900 --> 00:33:37,100 So that's what the rectangular coordinate system looks like. 590 00:33:37,100 --> 00:33:41,380 And now I should draw the other coordinate system that we have. 591 00:33:41,380 --> 00:33:47,900 Which is this guy here. 592 00:33:47,900 --> 00:33:49,610 Well, close enough. 593 00:33:49,610 --> 00:33:54,720 And these guys here. 594 00:33:54,720 --> 00:33:57,730 Kind of this bulls-eye or target operation. 595 00:33:57,730 --> 00:34:01,480 And this one is, say, theta = pi/2. 596 00:34:01,480 --> 00:34:03,870 This is theta = 0. 597 00:34:03,870 --> 00:34:07,710 This is theta = -pi/4. 598 00:34:07,710 --> 00:34:11,380 For instance, so I've just labeled for you three 599 00:34:11,380 --> 00:34:17,870 of the rays on this diagram. 600 00:34:17,870 --> 00:34:23,130 It's kind of like a radar screen. 601 00:34:23,130 --> 00:34:28,840 And then in pink, this is maybe r = 2, the radius 2. 602 00:34:28,840 --> 00:34:33,980 And inside is r = 1. 603 00:34:33,980 --> 00:34:38,090 So it's a different coordinate system for the plane. 604 00:34:38,090 --> 00:34:42,120 And again, the letter r represents measuring 605 00:34:42,120 --> 00:34:44,930 how far we are from the origin. 606 00:34:44,930 --> 00:34:47,060 The theta represents something about the angle, 607 00:34:47,060 --> 00:34:50,250 which ray we're on. 608 00:34:50,250 --> 00:34:52,260 And they're just two different variables. 609 00:34:52,260 --> 00:35:10,880 And this is a very different kind of coordinate system. 610 00:35:10,880 --> 00:35:15,391 OK so, our main job is just to get used to this. 611 00:35:15,391 --> 00:35:15,890 For now. 612 00:35:15,890 --> 00:35:18,350 You will be using this a lot in 18.02. 613 00:35:18,350 --> 00:35:20,570 It's very useful in physics. 614 00:35:20,570 --> 00:35:25,680 And our job is just to get started with it. 615 00:35:25,680 --> 00:35:29,990 And so, let's try a few examples here. 616 00:35:29,990 --> 00:35:31,220 Tons of examples. 617 00:35:31,220 --> 00:35:34,590 We'll start out very slow. 618 00:35:34,590 --> 00:35:41,860 If you have (x, y) = (1, -1), that's a point in the plane. 619 00:35:41,860 --> 00:35:44,380 I can draw that point. 620 00:35:44,380 --> 00:35:46,460 It's down here, right? 621 00:35:46,460 --> 00:35:50,630 This is -1 and this is 1, and here's my point, (1, -1). 622 00:35:50,630 --> 00:35:53,550 I can figure out what the representative is 623 00:35:53,550 --> 00:35:56,670 of this in polar coordinates. 624 00:35:56,670 --> 00:36:03,040 So in polar coordinates, there are actually 625 00:36:03,040 --> 00:36:05,130 a bunch of choices here. 626 00:36:05,130 --> 00:36:09,250 First of all, I'll tell you one choice. 627 00:36:09,250 --> 00:36:10,970 If I start with the angle horizontally, 628 00:36:10,970 --> 00:36:14,200 I wrap all the way around, that would 629 00:36:14,200 --> 00:36:19,350 be to this ray here-- Let's do it in green again. 630 00:36:19,350 --> 00:36:21,820 Alright, I labeled it actually as -pi/4, 631 00:36:21,820 --> 00:36:27,310 but another way of looking at it is that it's this angle here. 632 00:36:27,310 --> 00:36:31,440 So that would be r = square root of 2. 633 00:36:31,440 --> 00:36:34,210 Theta = 7pi/4. 634 00:36:38,150 --> 00:36:41,750 So that's one possibility of the angle and the distance. 635 00:36:41,750 --> 00:36:45,380 I know the distance is a square root of 2, that's not hard. 636 00:36:45,380 --> 00:36:47,930 Another way of looking at it is the way 637 00:36:47,930 --> 00:36:49,640 which was suggested when I labeled this 638 00:36:49,640 --> 00:36:51,230 with a negative angle. 639 00:36:51,230 --> 00:36:56,850 And that would be r = square root of 2, theta = -pi/4. 640 00:36:56,850 --> 00:36:58,370 And these are both legal. 641 00:36:58,370 --> 00:37:00,736 These are perfectly legal representatives. 642 00:37:00,736 --> 00:37:02,110 And that's what I meant by saying 643 00:37:02,110 --> 00:37:06,180 that these representations over here are somewhat ambiguous. 644 00:37:06,180 --> 00:37:08,900 There's more than one answer to this question, of what 645 00:37:08,900 --> 00:37:11,860 the polar representation is. 646 00:37:11,860 --> 00:37:17,190 A third possibility, which is even more dicey but also legal, 647 00:37:17,190 --> 00:37:21,890 is r equals minus square root of 2. 648 00:37:21,890 --> 00:37:25,360 Theta = 3pi/4. 649 00:37:25,360 --> 00:37:30,080 Now, what that corresponds to doing is going around to here. 650 00:37:30,080 --> 00:37:33,490 We're pointing out 3/4 pi direction. 651 00:37:33,490 --> 00:37:37,130 But then going negative square root of 2 distance. 652 00:37:37,130 --> 00:37:39,710 We're going backwards. 653 00:37:39,710 --> 00:37:42,250 So we're landing in the same place. 654 00:37:42,250 --> 00:37:44,380 So this is also legal. 655 00:37:44,380 --> 00:37:44,880 Yeah. 656 00:37:44,880 --> 00:37:51,324 STUDENT: [INAUDIBLE] 657 00:37:51,324 --> 00:37:53,240 PROFESSOR: The question is, don't the radiuses 658 00:37:53,240 --> 00:37:54,989 have to be positive because they represent 659 00:37:54,989 --> 00:37:56,620 a distance to the origin? 660 00:37:56,620 --> 00:38:00,620 The answer is I lied to you here. 661 00:38:00,620 --> 00:38:04,770 All of these things that I said are wrong, except for this. 662 00:38:04,770 --> 00:38:09,020 Which is the rule for what polar coordinates mean. 663 00:38:09,020 --> 00:38:21,170 So it's maybe plus or minus the distance, is what it is always. 664 00:38:21,170 --> 00:38:29,090 I try not to lie to you too much, but I do succeed. 665 00:38:29,090 --> 00:38:36,270 Now, let's do a little bit more practice here. 666 00:38:36,270 --> 00:38:38,330 There are some easy examples, which 667 00:38:38,330 --> 00:38:40,580 I will run through very quickly. r = a, 668 00:38:40,580 --> 00:38:44,100 we already know this is a circle. 669 00:38:44,100 --> 00:38:51,280 And the 3 theta equals a constant is a ray. 670 00:38:51,280 --> 00:38:54,820 However, this involves an implicit assumption, which 671 00:38:54,820 --> 00:38:57,360 I want to point out to you. 672 00:38:57,360 --> 00:38:59,040 So this is Example 3. 673 00:38:59,040 --> 00:39:01,060 Theta's equal to a constant is a ray. 674 00:39:01,060 --> 00:39:14,070 But this implicitly assumes 0 <= r < infinity. 675 00:39:14,070 --> 00:39:19,400 If you really wanted to allow minus infinity < r < infinity 676 00:39:19,400 --> 00:39:22,890 in this example, you would get a line. 677 00:39:22,890 --> 00:39:28,540 Gives the whole line. 678 00:39:28,540 --> 00:39:30,050 It gives everything behind. 679 00:39:30,050 --> 00:39:33,085 So you go out on some ray, you go backwards on that ray 680 00:39:33,085 --> 00:39:36,460 and you get the whole line through the origin, both ways. 681 00:39:36,460 --> 00:39:39,740 If you allow r going to minus infinity as well. 682 00:39:39,740 --> 00:39:42,310 So the typical conventions, so here 683 00:39:42,310 --> 00:39:49,680 are the typical conventions. 684 00:39:49,680 --> 00:39:53,140 And you will see people assume this without even telling you. 685 00:39:53,140 --> 00:39:55,340 So you need to watch out for it. 686 00:39:55,340 --> 00:39:57,450 The typical conventions are certainly this one, 687 00:39:57,450 --> 00:40:00,270 which is a nice thing to do. 688 00:40:00,270 --> 00:40:04,240 Pretty much all the time, although not all the time. 689 00:40:04,240 --> 00:40:05,360 Most of the time. 690 00:40:05,360 --> 00:40:11,950 And then you might have theta ranging from minus pi 691 00:40:11,950 --> 00:40:15,730 to pi, so in other words symmetric around 0. 692 00:40:15,730 --> 00:40:21,630 Or, another very popular choice is this one. 693 00:40:21,630 --> 00:40:25,890 Theta's >= 0 and strictly less than 2pi. 694 00:40:25,890 --> 00:40:29,660 So these are the two typical ranges 695 00:40:29,660 --> 00:40:33,930 in which all of these variables are chosen. 696 00:40:33,930 --> 00:40:34,900 But not always. 697 00:40:34,900 --> 00:40:43,210 You'll find that it's not consistent. 698 00:40:43,210 --> 00:40:46,010 As I said, our job is to get used to this. 699 00:40:46,010 --> 00:40:49,600 And I need to work up to some slightly more 700 00:40:49,600 --> 00:40:51,420 complicated examples. 701 00:40:51,420 --> 00:40:57,840 Some of which I'll give you on next Tuesday. 702 00:40:57,840 --> 00:41:05,780 But let's do a few more. 703 00:41:05,780 --> 00:41:10,820 So, I guess this is Example 4. 704 00:41:10,820 --> 00:41:14,980 Example 4, I'm going to take y = 1. 705 00:41:14,980 --> 00:41:20,650 That's awfully simple in rectangular coordinates. 706 00:41:20,650 --> 00:41:23,960 But interestingly, you might conceivably 707 00:41:23,960 --> 00:41:26,050 want to deal with it in polar coordinates. 708 00:41:26,050 --> 00:41:29,580 If you do, so here's how you make the translation. 709 00:41:29,580 --> 00:41:32,850 But this translation is not so terrible. 710 00:41:32,850 --> 00:41:39,080 What you do is, you plug in y = r sin(theta). 711 00:41:39,080 --> 00:41:40,710 That's all you have to do. 712 00:41:40,710 --> 00:41:42,760 And so that's going to be equal to 1. 713 00:41:42,760 --> 00:41:46,240 And that's going to give us our polar equation. 714 00:41:46,240 --> 00:41:50,330 The polar equation is r = 1 / sin(theta). 715 00:41:50,330 --> 00:41:54,360 There it is. 716 00:41:54,360 --> 00:41:58,120 And let's draw a picture of it. 717 00:41:58,120 --> 00:42:03,480 So here's a picture of the line y = 1. 718 00:42:03,480 --> 00:42:11,950 And now we see that if we take our rays going out from here, 719 00:42:11,950 --> 00:42:17,240 they collide with the line at various lengths. 720 00:42:17,240 --> 00:42:19,760 So if you take an angle, theta, here there'll 721 00:42:19,760 --> 00:42:21,364 be a distance r corresponding to that 722 00:42:21,364 --> 00:42:23,030 and you'll hit this in exactly one spot. 723 00:42:23,030 --> 00:42:26,600 For each theta you'll have a different radius. 724 00:42:26,600 --> 00:42:27,810 And it's a variable radius. 725 00:42:27,810 --> 00:42:30,740 It's given by this formula here. 726 00:42:30,740 --> 00:42:33,210 And so to trace this line out, you actually 727 00:42:33,210 --> 00:42:36,120 have to realize that there's one more thing involved. 728 00:42:36,120 --> 00:42:40,160 Which is the possible range of theta. 729 00:42:40,160 --> 00:42:41,730 Again, when you're doing integrations 730 00:42:41,730 --> 00:42:44,104 you're going to need to know those limits of integration. 731 00:42:44,104 --> 00:42:46,360 So you're going to need to know this. 732 00:42:46,360 --> 00:42:48,990 The range here goes from theta = 0, 733 00:42:48,990 --> 00:42:51,230 that's sort of when it's out at infinity. 734 00:42:51,230 --> 00:42:53,140 That's when the denominator is 0 here. 735 00:42:53,140 --> 00:42:55,800 And it goes all the way to pi. 736 00:42:55,800 --> 00:42:57,940 Swing around just one half-turn. 737 00:42:57,940 --> 00:43:03,610 So the range here is 0 < theta < pi. 738 00:43:03,610 --> 00:43:04,620 Yeah, question. 739 00:43:04,620 --> 00:43:09,676 STUDENT: [INAUDIBLE] 740 00:43:09,676 --> 00:43:11,050 PROFESSOR: The question is, is it 741 00:43:11,050 --> 00:43:13,940 typical to express r as a function of theta, 742 00:43:13,940 --> 00:43:16,550 or vice versa, or does it matter? 743 00:43:16,550 --> 00:43:19,790 The answer is that for the purposes of this course, 744 00:43:19,790 --> 00:43:24,420 we're almost always going to be writing things in this form. 745 00:43:24,420 --> 00:43:27,070 r as a function of theta. 746 00:43:27,070 --> 00:43:30,050 And you can do whatever you want. 747 00:43:30,050 --> 00:43:33,920 This turns out to be what we'll be doing in this course, 748 00:43:33,920 --> 00:43:37,040 exclusively. 749 00:43:37,040 --> 00:43:40,570 As you'll see when we get to other examples, 750 00:43:40,570 --> 00:43:42,160 it's the traditional sort of thing 751 00:43:42,160 --> 00:43:45,060 to do when you're thinking about observing a planet 752 00:43:45,060 --> 00:43:48,650 or something like that. 753 00:43:48,650 --> 00:43:52,930 You see the angle, and then you guess far away it is. 754 00:43:52,930 --> 00:43:55,600 But it's not necessary. 755 00:43:55,600 --> 00:43:58,940 The formulas are often easier this way. 756 00:43:58,940 --> 00:44:00,370 For the examples that we have. 757 00:44:00,370 --> 00:44:02,610 Because it's usually a trig function of theta. 758 00:44:02,610 --> 00:44:05,110 Whereas the other way, it would be an inverse trig function. 759 00:44:05,110 --> 00:44:08,930 So it's an uglier expression. 760 00:44:08,930 --> 00:44:10,540 As you can see. 761 00:44:10,540 --> 00:44:12,860 The real reason is that we choose this thing that's 762 00:44:12,860 --> 00:44:19,410 easier to deal with. 763 00:44:19,410 --> 00:44:22,200 So now let me give you a slightly more complicated 764 00:44:22,200 --> 00:44:24,410 example of the same type. 765 00:44:24,410 --> 00:44:28,930 Where we use a shortcut. 766 00:44:28,930 --> 00:44:31,680 This is a standard example. 767 00:44:31,680 --> 00:44:33,960 And it comes up a lot. 768 00:44:33,960 --> 00:44:40,730 And so this is an off-center circle. 769 00:44:40,730 --> 00:44:44,000 A circle is really easy to describe, but not 770 00:44:44,000 --> 00:44:54,170 necessarily if the center is on the rim of the circle. 771 00:44:54,170 --> 00:44:56,550 So that's a different problem. 772 00:44:56,550 --> 00:44:59,990 And let's do this with a circle of radius a. 773 00:44:59,990 --> 00:45:06,120 So this is the point (a, 0) and this is (2a, 0). 774 00:45:06,120 --> 00:45:08,550 And actually, if you know these two numbers, 775 00:45:08,550 --> 00:45:11,080 you'll be able to remember the result of this calculation. 776 00:45:11,080 --> 00:45:13,780 Which you'll do about five or six times and then finally 777 00:45:13,780 --> 00:45:17,310 you'll memorize it during 18.02 when you will need it a lot. 778 00:45:17,310 --> 00:45:21,220 So this is a standard calculation here. 779 00:45:21,220 --> 00:45:24,350 So the starting place is the rectangular equation. 780 00:45:24,350 --> 00:45:27,170 And we're going to pass to the polar representation. 781 00:45:27,170 --> 00:45:33,550 The rectangular representation is (x-a)^2 + y^2 = a^2. 782 00:45:33,550 --> 00:45:40,290 So this is a circle centered at (a, 0) of radius a. 783 00:45:40,290 --> 00:45:44,110 And now, if you like, the slow way of doing this 784 00:45:44,110 --> 00:45:50,145 would be to plug in x = r cos(theta), y = r sin(theta). 785 00:45:50,145 --> 00:45:51,520 The way I did in this first step. 786 00:45:51,520 --> 00:45:53,500 And that works perfectly well. 787 00:45:53,500 --> 00:45:56,980 But I'm going to do it more quickly than that. 788 00:45:56,980 --> 00:46:00,070 Because I can sort of see in advance how it's going to work. 789 00:46:00,070 --> 00:46:09,810 I'm just going to expand this out. 790 00:46:09,810 --> 00:46:13,160 And now I see the a^2's cancel. 791 00:46:13,160 --> 00:46:17,120 And not only that, but x^2 + y^2 = r^2. 792 00:46:17,120 --> 00:46:19,670 So this becomes r^2. 793 00:46:19,670 --> 00:46:28,590 That's x^2 + y^2 - 2ax = 0. 794 00:46:28,590 --> 00:46:32,360 The r came from the fact that r^2 = x^2 + y^2. 795 00:46:36,100 --> 00:46:37,890 So I'm doing this the rapid way. 796 00:46:37,890 --> 00:46:40,260 You can do it by plugging in, as I said. 797 00:46:40,260 --> 00:46:43,900 r equals-- So now that I've simplified it, 798 00:46:43,900 --> 00:46:45,720 I am going to use that procedure. 799 00:46:45,720 --> 00:46:47,570 I'm going to plug in. 800 00:46:47,570 --> 00:46:57,120 So here I have r^2 - 2ar cos(theta) = 0. 801 00:46:57,120 --> 00:47:00,146 I just plugged in for x. 802 00:47:00,146 --> 00:47:02,270 As I said, I could have done that at the beginning. 803 00:47:02,270 --> 00:47:06,430 I just simplified first. 804 00:47:06,430 --> 00:47:11,780 And now, this is the same thing as r^2 = 2ar cos(theta). 805 00:47:11,780 --> 00:47:13,530 And we're almost done. 806 00:47:13,530 --> 00:47:19,230 There's a boring part of this equation, which is r = 0. 807 00:47:19,230 --> 00:47:21,530 And then there's, if I divide by r, 808 00:47:21,530 --> 00:47:23,430 there's the interesting part of the equation. 809 00:47:23,430 --> 00:47:25,830 Which is this. 810 00:47:25,830 --> 00:47:28,810 So this is or r = 0. 811 00:47:28,810 --> 00:47:33,690 Which is already included in that equation anyway. 812 00:47:33,690 --> 00:47:36,890 So I'm allowed to divide by r because in the case of r = 0, 813 00:47:36,890 --> 00:47:39,781 this is represented anyway. 814 00:47:39,781 --> 00:47:40,280 Question. 815 00:47:40,280 --> 00:47:44,390 STUDENT: [INAUDIBLE] 816 00:47:44,390 --> 00:47:46,270 PROFESSOR: r = 0 is just one case. 817 00:47:46,270 --> 00:47:48,380 That is, it's the union of these two. 818 00:47:48,380 --> 00:47:49,550 It's both. 819 00:47:49,550 --> 00:47:50,670 Both are possible. 820 00:47:50,670 --> 00:47:53,270 So r = 0 is one point on it. 821 00:47:53,270 --> 00:47:56,150 And this is all of it. 822 00:47:56,150 --> 00:48:01,230 So we can just ignore this. 823 00:48:01,230 --> 00:48:04,500 So now I want to say one more important thing. 824 00:48:04,500 --> 00:48:06,600 You need to understand the range of this. 825 00:48:06,600 --> 00:48:10,840 So wait a second and we're going to figure out the range here. 826 00:48:10,840 --> 00:48:13,710 The range is very important, because otherwise you'll 827 00:48:13,710 --> 00:48:18,280 never be able to integrate using this representation here. 828 00:48:18,280 --> 00:48:19,840 So this is the representation. 829 00:48:19,840 --> 00:48:25,190 But notice when theta = 0, we're out here at 2a. 830 00:48:25,190 --> 00:48:26,780 That's consistent, and that's actually 831 00:48:26,780 --> 00:48:29,020 how you remember this factor 2a here. 832 00:48:29,020 --> 00:48:31,570 Because if you remember this picture and where you land when 833 00:48:31,570 --> 00:48:34,830 theta = 0. 834 00:48:34,830 --> 00:48:36,370 So that's the theta = 0 part. 835 00:48:36,370 --> 00:48:39,440 But now as I tip up like this, you 836 00:48:39,440 --> 00:48:43,780 see that when we get to vertical, we're done. 837 00:48:43,780 --> 00:48:44,630 With the circle. 838 00:48:44,630 --> 00:48:46,463 It's gotten shorter and shorter and shorter, 839 00:48:46,463 --> 00:48:49,020 and at theta = pi/2, we're down at 0. 840 00:48:49,020 --> 00:48:51,720 Because that's cos(pi/2) = 0. 841 00:48:51,720 --> 00:48:53,770 So it swings up like this. 842 00:48:53,770 --> 00:48:55,400 And it gets up to pi/2. 843 00:48:55,400 --> 00:48:57,110 Similarly, we swing down like this. 844 00:48:57,110 --> 00:48:59,000 And then we're done. 845 00:48:59,000 --> 00:49:04,510 So the range is -pi/2 < theta < pi/2. 846 00:49:04,510 --> 00:49:06,650 Or, if you want to throw in the r = 0 case, 847 00:49:06,650 --> 00:49:08,700 you can throw in this, this is repeating, 848 00:49:08,700 --> 00:49:11,200 if you like, at the ends. 849 00:49:11,200 --> 00:49:14,100 So this is the range of this circle. 850 00:49:14,100 --> 00:49:17,150 And let's see. 851 00:49:17,150 --> 00:49:21,300 Next time we'll figure out area in polar coordinates.