1 00:00:00 --> 00:00:06 2 00:00:01 --> 00:00:07 I usually like to start a lecture with something new, 3 00:00:04 --> 00:00:10 but this time I'm going to make an exception, 4 00:00:07 --> 00:00:13 and start with the finishing up on Friday because it involves a 5 00:00:11 --> 00:00:17 little more practice with complex numbers. 6 00:00:14 --> 00:00:20 I think that's what a large number of you are still fairly 7 00:00:18 --> 00:00:24 weak in. So, to briefly remind you, 8 00:00:20 --> 00:00:26 it will be sort of self-contained, 9 00:00:23 --> 00:00:29 but still, it will use complex numbers. 10 00:00:25 --> 00:00:31 And, I think it's a good way to start today. 11 00:00:30 --> 00:00:36 So, remember, the basic problem was to solve 12 00:00:33 --> 00:00:39 something with, where the input was sinusoidal 13 00:00:38 --> 00:00:44 in particular. The k was on both sides, 14 00:00:41 --> 00:00:47 and the input looked like cosine omega t. 15 00:00:46 --> 00:00:52 And, the plan of the solution consisted of transporting the 16 00:00:52 --> 00:00:58 problem to the complex domain. So, you look for a complex 17 00:00:57 --> 00:01:03 solution, and you complexify the right hand side of the equation, 18 00:01:02 --> 00:01:08 as well. So, cosine omega t 19 00:01:07 --> 00:01:13 becomes the real part of this complex function. 20 00:01:11 --> 00:01:17 The reason for doing that, remember, was because it's 21 00:01:15 --> 00:01:21 easier to handle when you solve linear equations. 22 00:01:19 --> 00:01:25 It's much easier to handle exponentials on the right-hand 23 00:01:23 --> 00:01:29 side than it is to handle sines and cosines because exponentials 24 00:01:28 --> 00:01:34 are so easy to integrate when you multiply them by other 25 00:01:33 --> 00:01:39 exponentials. So, the result was, 26 00:01:35 --> 00:01:41 after doing that, y tilda turned out to be one, 27 00:01:39 --> 00:01:45 after I scale the coefficient, one over one plus omega over k 28 00:01:44 --> 00:01:50 And then, the rest was e to the 29 00:01:51 --> 00:01:57 i times (omega t minus phi), 30 00:01:56 --> 00:02:02 where phi had a certain meaning. 31 00:02:00 --> 00:02:06 It was the arc tangent of a, it was a phase lag. 32 00:02:06 --> 00:02:12 And, this was then, I had to take the real part of 33 00:02:09 --> 00:02:15 this to get the final answer, which came out to be something 34 00:02:13 --> 00:02:19 like one over the square root of one plus the amplitude one omega 35 00:02:18 --> 00:02:24 / k squared, and then the rest was cosine omega t plus minus 36 00:02:22 --> 00:02:28 phi. 37 00:02:26 --> 00:02:32 It's easier to see that that part is the real part of this; 38 00:02:30 --> 00:02:36 the problem is, of course you have to convert 39 00:02:33 --> 00:02:39 this. Sorry, this should be i omega 40 00:02:37 --> 00:02:43 t, in which case you don't need the parentheses, 41 00:02:41 --> 00:02:47 either. So, the problem was to use the 42 00:02:45 --> 00:02:51 polar representation of this complex number to convert it 43 00:02:50 --> 00:02:56 into something whose amplitude was this, and whose angle was 44 00:02:55 --> 00:03:01 minus phi. Now, that's what we call the 45 00:02:58 --> 00:03:04 polar method, going polar. 46 00:03:02 --> 00:03:08 I'd like, now, for the first few minutes of 47 00:03:04 --> 00:03:10 the period, to talk about the other method, 48 00:03:07 --> 00:03:13 the Cartesian method. I think for a long while, 49 00:03:10 --> 00:03:16 many of you will be more comfortable with that anyway. 50 00:03:14 --> 00:03:20 Although, one of the objects of the course should be to get you 51 00:03:18 --> 00:03:24 equally comfortable with the polar representation of complex 52 00:03:22 --> 00:03:28 numbers. So, if we try to do the same 53 00:03:25 --> 00:03:31 thing going Cartesian, what's going to happen? 54 00:03:28 --> 00:03:34 Well, I guess the same point here. 55 00:03:32 --> 00:03:38 So, the starting point is still y tilde equals one over, 56 00:03:37 --> 00:03:43 sorry, this should have an i here, one plus i times omega 57 00:03:42 --> 00:03:48 over k, e to the i omega t. 58 00:03:48 --> 00:03:54 But now, what you're going to 59 00:03:52 --> 00:03:58 do is turn this into its Cartesian, turn both of these 60 00:03:57 --> 00:04:03 into their Cartesian representations as a plus ib. 61 00:04:02 --> 00:04:08 So, if you do that Cartesianly, 62 00:04:06 --> 00:04:12 of course, what you have to do is the standard thing about 63 00:04:09 --> 00:04:15 dividing complex numbers or taking the reciprocals that I 64 00:04:13 --> 00:04:19 told you at the very beginning of complex numbers. 65 00:04:16 --> 00:04:22 You multiply the top and bottom by the complex conjugate of this 66 00:04:21 --> 00:04:27 in order to make the bottom real. 67 00:04:23 --> 00:04:29 So, what does this become? This becomes one minus i times 68 00:04:27 --> 00:04:33 omega over k divided by the product of this in its complex 69 00:04:30 --> 00:04:36 conjugate, which is the real number, one plus omega over k 70 00:04:34 --> 00:04:40 squared 71 00:04:39 --> 00:04:45 So, I've now converted this to the a plus bi form. 72 00:04:43 --> 00:04:49 I have also to convert the right-hand side to the a plus bi 73 00:04:47 --> 00:04:53 form. So, it will look like cosine 74 00:04:49 --> 00:04:55 omega t plus i sine omega t. 75 00:04:53 --> 00:04:59 Having done that, 76 00:04:55 --> 00:05:01 I take the last step, which is to take the real part 77 00:04:59 --> 00:05:05 of that. Remember, the reason I want the 78 00:05:01 --> 00:05:07 real part is because this input was the real part of the complex 79 00:05:06 --> 00:05:12 input. So, once you've got the complex 80 00:05:10 --> 00:05:16 solution, you have to take its real part to go back into the 81 00:05:14 --> 00:05:20 domain you started with, of real numbers, 82 00:05:17 --> 00:05:23 from the domain of complex numbers. 83 00:05:19 --> 00:05:25 So, I want the real part is going to be, the real part of 84 00:05:23 --> 00:05:29 that is, first of all, there's a factor out in front. 85 00:05:27 --> 00:05:33 That's entirely real. Let's put that out in front, 86 00:05:32 --> 00:05:38 so doesn't bother us particularly. 87 00:05:34 --> 00:05:40 And now, I need the product of this complex number and that 88 00:05:39 --> 00:05:45 complex number. But, I only want the real part 89 00:05:42 --> 00:05:48 of it. So, I'm not going to multiply 90 00:05:45 --> 00:05:51 it out and get four terms. I'm just going to look at the 91 00:05:49 --> 00:05:55 two terms that I do want. I don't want the others. 92 00:05:53 --> 00:05:59 All right, the real part is cosine omega t, 93 00:05:58 --> 00:06:04 from the product of this and that. 94 00:06:02 --> 00:06:08 And, the rest of the real part will be the product of the two i 95 00:06:07 --> 00:06:13 terms. But, it's i times negative i, 96 00:06:11 --> 00:06:17 which makes one. And therefore, 97 00:06:13 --> 00:06:19 it's omega over k times sine omega t. 98 00:06:18 --> 00:06:24 Now, that's the answer. 99 00:06:22 --> 00:06:28 And that's the answer, too; they must be equal, 100 00:06:26 --> 00:06:32 unless there's a contradiction in mathematics. 101 00:06:32 --> 00:06:38 But, it's extremely important. And that's the other reason why 102 00:06:36 --> 00:06:42 I'm giving you this, that you learn in this course 103 00:06:40 --> 00:06:46 to be able to convert quickly and automatically things that 104 00:06:44 --> 00:06:50 look like this into things that look like that. 105 00:06:47 --> 00:06:53 And, that's done by means of a basic formula, 106 00:06:50 --> 00:06:56 which occurs at the end of the notes for reference, 107 00:06:54 --> 00:07:00 as I optimistically say, although I think for a lot of 108 00:06:58 --> 00:07:04 you will not be referenced, stuff in the category of, 109 00:07:02 --> 00:07:08 yeah, I think I've vaguely seen that somewhere. 110 00:07:07 --> 00:07:13 But, well, we never used it for anything. 111 00:07:10 --> 00:07:16 Okay, you're going to use it all term. 112 00:07:12 --> 00:07:18 So, the formula is, the famous trigonometric 113 00:07:16 --> 00:07:22 identity, which is, so, the problem is to convert 114 00:07:19 --> 00:07:25 this into the other guy. And, the thing which is going 115 00:07:24 --> 00:07:30 to do that, enable one to combine the sine and the cosine 116 00:07:28 --> 00:07:34 terms, is the famous formula that a times the cosine, 117 00:07:32 --> 00:07:38 I'm going to use theta to make it as neutral as possible, 118 00:07:36 --> 00:07:42 -- -- so, theta you can think of 119 00:07:40 --> 00:07:46 as being replaced by omega t in this particular application of 120 00:07:44 --> 00:07:50 the formula. But, I'll just use a general 121 00:07:47 --> 00:07:53 angle theta, which doesn't suggest anything in particular. 122 00:07:51 --> 00:07:57 So, the problem is, you have something which is a 123 00:07:54 --> 00:08:00 combination with real coefficients of cosine and sine, 124 00:07:58 --> 00:08:04 and the important thing is that these numbers be the same. 125 00:08:03 --> 00:08:09 In practice, that means that the omega t, 126 00:08:06 --> 00:08:12 you're not allowed to have omega one t here, 127 00:08:10 --> 00:08:16 and some other frequency, omega two t here. 128 00:08:14 --> 00:08:20 That would correspond to using theta one here, 129 00:08:17 --> 00:08:23 and theta two here. And, though there is a formula 130 00:08:21 --> 00:08:27 for combining that, nobody remembers it, 131 00:08:24 --> 00:08:30 and it's, in general, less universally useful than 132 00:08:28 --> 00:08:34 the first. If you're going to memorize a 133 00:08:32 --> 00:08:38 formula, and learn this one, it's best to start with the 134 00:08:38 --> 00:08:44 ones where the two are equal. That's the basic formula. 135 00:08:44 --> 00:08:50 The others are variations of it, but there is a sizable 136 00:08:49 --> 00:08:55 variations. All right, so the answer is 137 00:08:53 --> 00:08:59 that this is equal to some other constant, real constant, 138 00:08:59 --> 00:09:05 times the cosine of theta minus phi. 139 00:09:06 --> 00:09:12 Of course, most people remember this vaguely. 140 00:09:09 --> 00:09:15 What they don't remember is what the c and the phi are, 141 00:09:14 --> 00:09:20 how to calculate them. I don't suggest you memorize 142 00:09:18 --> 00:09:24 the formulas for them. You can if you wish. 143 00:09:22 --> 00:09:28 Instead, memorize the picture, which is much easier. 144 00:09:26 --> 00:09:32 Memorize that a and b are the two sides of a right triangle. 145 00:09:31 --> 00:09:37 Phi is the angle opposite the b side, and c is the length of the 146 00:09:36 --> 00:09:42 hypotenuse. Okay, that's worth putting up. 147 00:09:42 --> 00:09:48 I think that's a pink formula. It's even worth two of those, 148 00:09:48 --> 00:09:54 but I will thrift. Now, let's apply it to this 149 00:09:52 --> 00:09:58 case to see that it gives the right answer. 150 00:09:57 --> 00:10:03 So, to use this formula, how I use it? 151 00:10:02 --> 00:10:08 Well, I should take, I will reproduce the left-hand 152 00:10:07 --> 00:10:13 side. So that part, 153 00:10:09 --> 00:10:15 I just copy. And, how about the right? 154 00:10:13 --> 00:10:19 Well, the amplitude, it's combined into a single 155 00:10:19 --> 00:10:25 cosine term whose amplitude is, well, the two sides of the 156 00:10:25 --> 00:10:31 right triangle are one, and omega over k. 157 00:10:33 --> 00:10:39 The hypotenuse in that case is going to be, well, 158 00:10:36 --> 00:10:42 why don't we write it here? So, we have one, 159 00:10:39 --> 00:10:45 and omega over k. And, here's phi. 160 00:10:43 --> 00:10:49 So, the hypotenuse is going to be the square root of one plus 161 00:10:47 --> 00:10:53 omega over k squared. 162 00:10:51 --> 00:10:57 And, that's going to be multiplied by the cosine of 163 00:10:55 --> 00:11:01 omega t minus this phase lag angle phi. 164 00:10:59 --> 00:11:05 You can write, 165 00:11:02 --> 00:11:08 if you wish, phi equals the arc tangent, 166 00:11:05 --> 00:11:11 but you are not learning a lot by that. 167 00:11:09 --> 00:11:15 Phi is the arc tangent of omega over k. 168 00:11:15 --> 00:11:21 That's okay, 169 00:11:16 --> 00:11:22 but it's true. But, notice there's 170 00:11:19 --> 00:11:25 cancellation now. This over that is equal to 171 00:11:23 --> 00:11:29 what? Well, it's equal to this. 172 00:11:28 --> 00:11:34 And, so when we get in this way, by combining these two 173 00:11:31 --> 00:11:37 factors, one gets exactly the same formula that we got before. 174 00:11:36 --> 00:11:42 So, as you can see, in some sense, 175 00:11:38 --> 00:11:44 there's not, if you can remember this 176 00:11:41 --> 00:11:47 trigonometric identity, there's not a lot of difference 177 00:11:45 --> 00:11:51 between the two methods except that this one requires this 178 00:11:49 --> 00:11:55 extra step. The answer will come out in 179 00:11:51 --> 00:11:57 this form, and you then, to see what it really looks 180 00:11:55 --> 00:12:01 like, really have to convert it to this form, 181 00:11:58 --> 00:12:04 the form in which you can see what the phase lag and the 182 00:12:02 --> 00:12:08 amplitude is. It's amazing how many people 183 00:12:07 --> 00:12:13 who should know, this includes working 184 00:12:10 --> 00:12:16 mathematicians, theoretical mathematicians, 185 00:12:13 --> 00:12:19 includes even possibly the authors of your textbooks. 186 00:12:17 --> 00:12:23 I'm not sure, but I've caught them in this, 187 00:12:20 --> 00:12:26 too, who in this form, everybody remembers that it's 188 00:12:24 --> 00:12:30 something like that. Unfortunately, 189 00:12:27 --> 00:12:33 when it occurs as the answer in an answer book, 190 00:12:31 --> 00:12:37 the numbers are some colossal mess here plus some colossal 191 00:12:35 --> 00:12:41 mess here. And theta is, 192 00:12:39 --> 00:12:45 again, a real mess, involving roots and some cube 193 00:12:44 --> 00:12:50 roots, and whatnot. The only thing is, 194 00:12:48 --> 00:12:54 these two are the same real mess. 195 00:12:52 --> 00:12:58 That amounts to just another pure oscillation with the same 196 00:12:58 --> 00:13:04 frequency as the old guy, and with the amplitude changed, 197 00:13:05 --> 00:13:11 and with a phase shift, move to the right or left. 198 00:13:12 --> 00:13:18 So, this is no more general than that. 199 00:13:14 --> 00:13:20 Notice they both have two parameters in them, 200 00:13:18 --> 00:13:24 these two coefficients. This one has the two parameters 201 00:13:22 --> 00:13:28 in an altered form. Okay, well, I wanted, 202 00:13:26 --> 00:13:32 because of the importance of this formula, 203 00:13:29 --> 00:13:35 I wanted to take a couple of minutes out for a proof of the 204 00:13:34 --> 00:13:40 formula, -- 205 00:13:36 --> 00:13:42 206 00:13:42 --> 00:13:48 -- just to give you chance to stare at it a little more now. 207 00:13:46 --> 00:13:52 There are three proofs I know. I'm sure there are 27. 208 00:13:50 --> 00:13:56 The Pythagorean theorem now has several hundred. 209 00:13:53 --> 00:13:59 But, there are three basic proofs. 210 00:13:56 --> 00:14:02 There is the one I will not give you, I'll call the high 211 00:14:00 --> 00:14:06 school proof, which is the only one one 212 00:14:03 --> 00:14:09 normally finds in books, physics textbooks or other 213 00:14:07 --> 00:14:13 textbooks. The high school proof takes the 214 00:14:11 --> 00:14:17 right-hand side, applies the formula for the 215 00:14:14 --> 00:14:20 cosine of the difference of two angles, which it assumes you had 216 00:14:19 --> 00:14:25 in trigonometry, and then converts it into this. 217 00:14:22 --> 00:14:28 It shows you that once you've done that, that a turns out to 218 00:14:27 --> 00:14:33 be c cosine phi and b, the number b is c sine 219 00:14:31 --> 00:14:37 phi, and therefore it identifies the 220 00:14:35 --> 00:14:41 two sides. Now, the thing that's of course 221 00:14:39 --> 00:14:45 correct and it's the simplest possible argument, 222 00:14:42 --> 00:14:48 the thing that's no good about it is that the direction at 223 00:14:47 --> 00:14:53 which it goes is from here to here. 224 00:14:49 --> 00:14:55 Well, everybody knew that. If I gave you this and told 225 00:14:53 --> 00:14:59 you, write it out in terms of cosine and sine, 226 00:14:56 --> 00:15:02 I would assume it dearly hope that practically all of you can 227 00:15:01 --> 00:15:07 do that. Unfortunately, 228 00:15:03 --> 00:15:09 when you want to use the formula, it's this way you want 229 00:15:07 --> 00:15:13 to use it in the opposite direction. 230 00:15:09 --> 00:15:15 You are starting with this, and want to convert it to that. 231 00:15:13 --> 00:15:19 Now, the proof, therefore, will not be of much 232 00:15:15 --> 00:15:21 help. It requires you to go in the 233 00:15:17 --> 00:15:23 backwards direction, and match up coefficients. 234 00:15:20 --> 00:15:26 It's much better to go forwards. 235 00:15:22 --> 00:15:28 Now, there are two proofs that go forwards. 236 00:15:25 --> 00:15:31 There's the 18.02 proof. Since I didn't teach most of 237 00:15:28 --> 00:15:34 you 18.02, I can't be sure you had it. 238 00:15:32 --> 00:15:38 So, I'll spend one minute giving it to you. 239 00:15:36 --> 00:15:42 What is the 18.02 proof? It is the following picture. 240 00:15:42 --> 00:15:48 I think this requires deep colored chalk. 241 00:15:46 --> 00:15:52 This is going to be pretty heavy. 242 00:15:50 --> 00:15:56 All right, first of all, the a and the b are the given. 243 00:15:55 --> 00:16:01 So, I'm going to put in that vector. 244 00:16:01 --> 00:16:07 So, there is the vector whose sides are, whose components are 245 00:16:05 --> 00:16:11 a and b. I'll write it without the i and 246 00:16:08 --> 00:16:14 j. I hope you had from Jerison 247 00:16:11 --> 00:16:17 that form for the vector, if you don't like that, 248 00:16:14 --> 00:16:20 write ai plus bj, okay? 249 00:16:17 --> 00:16:23 Now, there's another vector lurking around. 250 00:16:20 --> 00:16:26 It's the unit vector whose, I'll write it this way, 251 00:16:24 --> 00:16:30 u because it's a unit vector, and theta to indicate that it's 252 00:16:29 --> 00:16:35 angle is theta. Now, the reason for doing that 253 00:16:34 --> 00:16:40 is because you see that the left-hand side is a dot product 254 00:16:38 --> 00:16:44 of two vectors. The left-hand side of the 255 00:16:41 --> 00:16:47 identity is the dot product of the vector 00:16:51 b> with the vector whose components are cosine theta and 257 00:16:49 --> 00:16:55 sine theta. 258 00:16:52 --> 00:16:58 That's what I'm calling this unit vector. 259 00:16:55 --> 00:17:01 It's a unit vector because cosine squared plus sine squared 260 00:16:59 --> 00:17:05 is one. 261 00:17:04 --> 00:17:10 Now, all this formula is, is saying that scalar product, 262 00:17:08 --> 00:17:14 the dot product of those two vectors, can be evaluated if you 263 00:17:13 --> 00:17:19 know their components by the left-hand side of the formula. 264 00:17:18 --> 00:17:24 And, if you don't know their components, it can be evaluated 265 00:17:24 --> 00:17:30 in another way, the geometric evaluation, 266 00:17:27 --> 00:17:33 which goes, what is it? It's a magnitude of one, 267 00:17:31 --> 00:17:37 times the magnitude of the other, times the cosine of the 268 00:17:36 --> 00:17:42 included angle. Now, what's the included angle? 269 00:17:42 --> 00:17:48 Well, theta is this angle from the horizontal to that unit 270 00:17:49 --> 00:17:55 vector. The angle phi is this angle, 271 00:17:54 --> 00:18:00 from this picture here. And therefore, 272 00:17:58 --> 00:18:04 the included angle between (u)theta and my pink vector is 273 00:18:05 --> 00:18:11 theta minus phi. That's the formula. 274 00:18:12 --> 00:18:18 It comes from two ways of calculating the scalar product 275 00:18:16 --> 00:18:22 of the vector whose coefficients are, and the unit vector 276 00:18:21 --> 00:18:27 whose components are cosine theta and sine theta. 277 00:18:25 --> 00:18:31 All right, well, 278 00:18:28 --> 00:18:34 you should, that was 18.02. 279 00:18:35 --> 00:18:41 There must be an 18.03 proof also. Yes. 280 00:18:36 --> 00:18:42 What's the 18.03 proof? The 18.03 proof uses complex 281 00:18:43 --> 00:18:49 numbers. It says, look, 282 00:18:46 --> 00:18:52 take the left side. Instead of viewing it as the 283 00:18:53 --> 00:18:59 dot product of two vectors, there's another way. 284 00:19:02 --> 00:19:08 You can think of it as the part of the products of two complex 285 00:19:06 --> 00:19:12 numbers. So, the 18.03 argument, 286 00:19:09 --> 00:19:15 really, the complex number argument says, 287 00:19:12 --> 00:19:18 look, multiply together a minus bi and the complex 288 00:19:17 --> 00:19:23 number cosine theta plus i sine theta. 289 00:19:21 --> 00:19:27 There are different ways of 290 00:19:24 --> 00:19:30 explaining why I want to put the minus i there instead of i. 291 00:19:28 --> 00:19:34 But, the simplest is because I want, when I take the real part, 292 00:19:33 --> 00:19:39 to get the left-hand side. I will. 293 00:19:37 --> 00:19:43 If I take the real part of this, I'm going to get a cosine 294 00:19:42 --> 00:19:48 theta plus b sine theta 295 00:19:46 --> 00:19:52 because of negative i and i make one, 296 00:19:51 --> 00:19:57 multiplied together. All right, that's the left-hand 297 00:19:55 --> 00:20:01 side. And now, the right-hand side, 298 00:19:58 --> 00:20:04 I'm going to use polar representation instead. 299 00:20:03 --> 00:20:09 What's the polar representation of this guy? 300 00:20:06 --> 00:20:12 Well, if has the angle theta, 301 00:20:09 --> 00:20:15 then a negative b, a minus bi goes down 302 00:20:13 --> 00:20:19 below. It has the angle minus phi. 303 00:20:16 --> 00:20:22 So, this is, has magnitude. 304 00:20:18 --> 00:20:24 It is polar representation. Its magnitude is a squared plus 305 00:20:23 --> 00:20:29 b squared, and its angle is negative phi, 306 00:20:27 --> 00:20:33 not positive phi because this a minus bi goes below 307 00:20:32 --> 00:20:38 the axis if a and b are positive. 308 00:20:36 --> 00:20:42 So, it's e to the minus i phi. 309 00:20:39 --> 00:20:45 That's the first guy. And, how about the second guy? 310 00:20:43 --> 00:20:49 Well, the second guy is e to the i theta. 311 00:20:47 --> 00:20:53 So, what's the product? It is a squared plus b squared, 312 00:20:51 --> 00:20:57 the square root, times e to the i times (theta 313 00:20:54 --> 00:21:00 minus phi). 314 00:20:58 --> 00:21:04 And now, what do I want? The real part of this, 315 00:21:01 --> 00:21:07 and I want the real part of this. 316 00:21:05 --> 00:21:11 So, let's just say take the real parts of both sides. 317 00:21:08 --> 00:21:14 If I take the real part of the left-hand side, 318 00:21:11 --> 00:21:17 I get a cosine theta plus b sine theta. 319 00:21:14 --> 00:21:20 If I take a real part of this 320 00:21:17 --> 00:21:23 side, I get square root of a squared plus b squared times e, 321 00:21:21 --> 00:21:27 times the cosine, that's the real part, 322 00:21:23 --> 00:21:29 right, of theta minus phi, which is just what it's 323 00:21:26 --> 00:21:32 supposed to be. 324 00:21:31 --> 00:21:37 Well, with three different arguments, I'm really pounding 325 00:21:35 --> 00:21:41 the table on this formula. But, I think there's something 326 00:21:40 --> 00:21:46 to be learned from at least two of them. 327 00:21:44 --> 00:21:50 And, you know, I'm still, for awhile, 328 00:21:47 --> 00:21:53 I will never miss an opportunity to bang complex 329 00:21:51 --> 00:21:57 numbers into your head because, in some sense, 330 00:21:55 --> 00:22:01 you have to reproduce the experience of the race. 331 00:22:01 --> 00:22:07 As I mentioned in the notes, it took mathematicians 300 or 332 00:22:04 --> 00:22:10 400 years to get used to complex numbers. 333 00:22:07 --> 00:22:13 So, if it takes you three or four weeks, that's not too bad. 334 00:22:11 --> 00:22:17 335 00:22:32 --> 00:22:38 Now, for the rest of the period I'd like to go back to the 336 00:22:36 --> 00:22:42 linear equations, and try to put into perspective 337 00:22:40 --> 00:22:46 and summarize, and tell you a couple of things 338 00:22:43 --> 00:22:49 which I had to leave out, but which are, 339 00:22:46 --> 00:22:52 in my view, extremely important. 340 00:22:48 --> 00:22:54 And, up to now, I don't want to leave you with 341 00:22:52 --> 00:22:58 any misapprehensions. So, I'm going to summarize this 342 00:22:56 --> 00:23:02 way, whereas last lecture I went from the most general equation 343 00:23:01 --> 00:23:07 to the most special. I'd like to just write them 344 00:23:06 --> 00:23:12 down in the reverse order, now. 345 00:23:09 --> 00:23:15 So, we are talking about basic linear equations. 346 00:23:13 --> 00:23:19 First order, of course, we haven't moved as 347 00:23:16 --> 00:23:22 a second order yet. So, the most special one, 348 00:23:20 --> 00:23:26 and the one we talked about essentially all of the previous 349 00:23:25 --> 00:23:31 two times, or last Friday, anyway, was the equation where 350 00:23:30 --> 00:23:36 the k, the coefficient of y, is constant, 351 00:23:34 --> 00:23:40 and where you also get it on the right-hand side quite 352 00:23:39 --> 00:23:45 providentially. So, this is the most special 353 00:23:44 --> 00:23:50 form, and it's the one which governed what I will call the 354 00:23:48 --> 00:23:54 temperature-concentration model, or if you want to be grown up, 355 00:23:53 --> 00:23:59 the conduction-diffusion model, conduction-diffusion which 356 00:23:57 --> 00:24:03 describes the processes, which the equation is modeling, 357 00:24:01 --> 00:24:07 whereas these simply described the variables of things, 358 00:24:05 --> 00:24:11 which you usually are trying to calculate when you use the 359 00:24:10 --> 00:24:16 equation. Now, there are a class of 360 00:24:13 --> 00:24:19 things where the thing is constant, but where the k does 361 00:24:17 --> 00:24:23 not appear naturally on the right hand side. 362 00:24:20 --> 00:24:26 And, you're going to encounter them pretty quickly in physics, 363 00:24:24 --> 00:24:30 for one place. So, I better not try to sweep 364 00:24:27 --> 00:24:33 those under the rug. Let's just call that q of t. 365 00:24:32 --> 00:24:38 And finally, there is the most general case, 366 00:24:36 --> 00:24:42 where you allow k to be non-constant. 367 00:24:40 --> 00:24:46 That's the one we began, when we talked about the linear 368 00:24:45 --> 00:24:51 equation. And you know how to solve it in 369 00:24:49 --> 00:24:55 general by a definite or an indefinite integral. 370 00:24:54 --> 00:25:00 Now, there's one other thing, which I want to talk about. 371 00:25:01 --> 00:25:07 I will do all these in a certain order. 372 00:25:03 --> 00:25:09 But, from the beginning, you should keep in mind that 373 00:25:07 --> 00:25:13 there's another between the first two cases. 374 00:25:10 --> 00:25:16 Between the first two cases, there's another extremely 375 00:25:14 --> 00:25:20 important distinction, and that is as to whether k is 376 00:25:18 --> 00:25:24 positive or not. Up to now, we've always had k 377 00:25:21 --> 00:25:27 positive. So, I'm going to put that here. 378 00:25:24 --> 00:25:30 So, it's understood when I write these, that k is positive. 379 00:25:30 --> 00:25:36 I want to talk about that, too. 380 00:25:32 --> 00:25:38 But, first things first. The first thing I wanted to do 381 00:25:36 --> 00:25:42 was to show you that this, the first case, 382 00:25:39 --> 00:25:45 the most special case, does not just apply to this. 383 00:25:43 --> 00:25:49 It applies to other things, too. 384 00:25:45 --> 00:25:51 Let me give you a mixing problem. 385 00:25:48 --> 00:25:54 The typical mixing problem gives another example. 386 00:25:51 --> 00:25:57 You've already done in recitation, and you did one for 387 00:25:55 --> 00:26:01 the problem set, the problem of the two rooms 388 00:25:59 --> 00:26:05 filled with smoke. But, let me do it just using 389 00:26:04 --> 00:26:10 letters, so that the ideas stand out a little more clearly, 390 00:26:08 --> 00:26:14 and you are not preoccupied with the numbers, 391 00:26:11 --> 00:26:17 and calculating with the numbers, and trying to get 392 00:26:15 --> 00:26:21 numerical examples. So, it's as simple as k sub 393 00:26:19 --> 00:26:25 mixing. It looks like this. 394 00:26:21 --> 00:26:27 You have a tank, a room, I don't know, 395 00:26:23 --> 00:26:29 where everything's getting mixed in. 396 00:26:26 --> 00:26:32 It has a certain volume, which I will call v. 397 00:26:31 --> 00:26:37 Something is flowing in, a gas or a liquid. 398 00:26:34 --> 00:26:40 And, r will be the flow rate, in some units. 399 00:26:38 --> 00:26:44 Now, since it can't pile up inside this sealed container, 400 00:26:44 --> 00:26:50 which I'm sure is full, the flow rate out must also be 401 00:26:49 --> 00:26:55 r. And, what we're interested in 402 00:26:51 --> 00:26:57 is the amount of salt. So, x, let's suppose these are 403 00:26:56 --> 00:27:02 fluid flows, and the dissolved substance that I'm talking about 404 00:27:02 --> 00:27:08 is not carbon monoxide, it's salt, any dissolved 405 00:27:06 --> 00:27:12 substance, some pollutant or whatever the problem calls for. 406 00:27:14 --> 00:27:20 Let's use salt. So, it's the amount of salt in 407 00:27:20 --> 00:27:26 the tank at time t. I'm interested in knowing how 408 00:27:28 --> 00:27:34 that varies with time. Now, there's nothing to be said 409 00:27:34 --> 00:27:40 about how it flows out. What flows out, 410 00:27:36 --> 00:27:42 of course, is what happens to be in the tank. 411 00:27:39 --> 00:27:45 But, I do have to say what flows in. 412 00:27:41 --> 00:27:47 Now, the only convenient way to describe the in-flow is in terms 413 00:27:46 --> 00:27:52 of its concentration. The salt will be dissolved in 414 00:27:49 --> 00:27:55 the in-flowing water, and so there will be a certain 415 00:27:52 --> 00:27:58 concentration. And, as you will see, 416 00:27:55 --> 00:28:01 for a secret reason, I'm going to give that the 417 00:27:58 --> 00:28:04 subscript e. So, e is the concentration of 418 00:28:02 --> 00:28:08 the incoming salt, in other words, 419 00:28:05 --> 00:28:11 in the fluid, how many grams are there per 420 00:28:09 --> 00:28:15 liter in the incoming fluid. That's the data. 421 00:28:13 --> 00:28:19 So, this is the data. r is part of the data. 422 00:28:17 --> 00:28:23 r is the flow rate. v is the volume. 423 00:28:20 --> 00:28:26 I think I won't bother writing that down, and the problem is to 424 00:28:25 --> 00:28:31 determine what happens to x of t. 425 00:28:30 --> 00:28:36 Now, I strongly recommend you not attempt to work directly 426 00:28:34 --> 00:28:40 with the concentrations unless you feel you have a really good 427 00:28:39 --> 00:28:45 physical feeling for concentrations. 428 00:28:42 --> 00:28:48 I strongly recommend you work with a variable that you are 429 00:28:46 --> 00:28:52 given, namely, the dependent variable, 430 00:28:49 --> 00:28:55 which is the amount of salt, grams. 431 00:28:52 --> 00:28:58 Well, because it's something you can physically think about. 432 00:28:57 --> 00:29:03 It's coming in, it's getting mixed up, 433 00:29:00 --> 00:29:06 and some of it is going out. So, the basic equation is going 434 00:29:08 --> 00:29:14 to be that the rate of change of salt in the tank is the rate of 435 00:29:18 --> 00:29:24 salt inflow, let me write salt inflow, minus the rate of salt 436 00:29:27 --> 00:29:33 outflow. Okay, at what rate is salt 437 00:29:33 --> 00:29:39 flowing in? Well, the flow rate is the flow 438 00:29:39 --> 00:29:45 rate of the liquid. I multiply the flow rate, 439 00:29:43 --> 00:29:49 1 L per minute times the concentration, 440 00:29:47 --> 00:29:53 3 g per liter. That means 3 g per minute. 441 00:29:51 --> 00:29:57 It's going to be, therefore, the product of the 442 00:29:56 --> 00:30:02 flow rate and the concentration, incoming concentration. 443 00:30:03 --> 00:30:09 How about the rate of the salt outflow? 444 00:30:05 --> 00:30:11 Well, again, the rate of the liquid outflow 445 00:30:08 --> 00:30:14 is r. And, what is the concentration 446 00:30:11 --> 00:30:17 of salt in the outflow? I must use, when you talk flow 447 00:30:15 --> 00:30:21 rates, the other factor must be the concentration, 448 00:30:18 --> 00:30:24 not the amount. So, what is the concentration 449 00:30:22 --> 00:30:28 in the outflow? Well, it's the amount of salt 450 00:30:25 --> 00:30:31 in the tank divided by its volume. 451 00:30:29 --> 00:30:35 So, the analog, the concentration here is x 452 00:30:32 --> 00:30:38 divided by v. Now, here's a typical messy 453 00:30:36 --> 00:30:42 equation, dx / dt, let's write it in the standard, 454 00:30:40 --> 00:30:46 linear form, plus r times x over v equals r 455 00:30:44 --> 00:30:50 times the given concentration, which is a function of time. 456 00:30:50 --> 00:30:56 Now, this is going to be some given function, 457 00:30:54 --> 00:31:00 and there will be no reason whatsoever why you can't solve 458 00:30:59 --> 00:31:05 it in that form. And, that's normally what you 459 00:31:04 --> 00:31:10 will do. Nonetheless, 460 00:31:05 --> 00:31:11 in trying to understand how it fits into this paradigm, 461 00:31:09 --> 00:31:15 which kind of equation is it? Well, clearly there's an 462 00:31:12 --> 00:31:18 awkwardness in that on the right-hand side, 463 00:31:15 --> 00:31:21 we have concentration, and on the left-hand side, 464 00:31:19 --> 00:31:25 we seem to have amounts. Now, the way to understand the 465 00:31:22 --> 00:31:28 equation as opposed to the way to solve it, well, 466 00:31:26 --> 00:31:32 it's a step on the way to solving it. 467 00:31:28 --> 00:31:34 But, I emphasize, you can and normally will solve 468 00:31:32 --> 00:31:38 it in exactly that form. But, to understand what's 469 00:31:36 --> 00:31:42 happening, it's better to express it in terms of 470 00:31:40 --> 00:31:46 concentration entirely, and that's why it's called the 471 00:31:43 --> 00:31:49 concentration, or the diffusion, 472 00:31:45 --> 00:31:51 concentration-diffusion equation. 473 00:31:48 --> 00:31:54 So, I'm going to convert this to concentrations. 474 00:31:51 --> 00:31:57 Now, there's no problem here. x over v is the concentration 475 00:31:55 --> 00:32:01 in the tank. And now, immediately, 476 00:31:57 --> 00:32:03 you see, hey, it looks like it's going to 477 00:32:00 --> 00:32:06 come out just in the first form. But, wait a minute. 478 00:32:06 --> 00:32:12 How about the x? How do I convert that? 479 00:32:10 --> 00:32:16 Well, what's the relation between x? 480 00:32:13 --> 00:32:19 So, if the concentration in the tank is equal to x over v, 481 00:32:19 --> 00:32:25 so the tank concentration, then x is equal to C times the 482 00:32:25 --> 00:32:31 constant, V, and dx / dt, therefore, will be c times dC / 483 00:32:31 --> 00:32:37 dt. You see that? 484 00:32:34 --> 00:32:40 Now, that's not in standard form. 485 00:32:38 --> 00:32:44 Let's put it in standard form. To put it in standard form, 486 00:32:44 --> 00:32:50 I see, now, that it's not r that's the critical quantity. 487 00:32:51 --> 00:32:57 It's r divided by v. So, it's dC / dt, 488 00:32:55 --> 00:33:01 C prime, plus r divided by v, -- 489 00:33:00 --> 00:33:06 -- I'm going to call that k, k C, no let's not, 490 00:33:04 --> 00:33:10 r divided by v is equal to r divided by v times Ce. 491 00:33:09 --> 00:33:15 That's the equation expressed 492 00:33:14 --> 00:33:20 in a form where the concentration is the dependent 493 00:33:18 --> 00:33:24 variable, rather than the amount of salt itself. 494 00:33:23 --> 00:33:29 And, you can see it falls exactly in this category. 495 00:33:27 --> 00:33:33 That means that I can talk about it. 496 00:33:31 --> 00:33:37 The natural way to talk about this equation is in terms of, 497 00:33:36 --> 00:33:42 the same way we talked about the temperature equation. 498 00:33:43 --> 00:33:49 I said concentration. I mean, that concentration has 499 00:33:47 --> 00:33:53 nothing to do with this concentration. 500 00:33:50 --> 00:33:56 This is the diffusion model, where salt solution outside, 501 00:33:54 --> 00:34:00 cell in the middle, salt diffusing through a 502 00:33:58 --> 00:34:04 semi-permeable membrane into that, uses Newton's law of 503 00:34:03 --> 00:34:09 diffusion, except he didn't do a law of diffusion. 504 00:34:08 --> 00:34:14 But, he is sticky. His name is attached to 505 00:34:10 --> 00:34:16 everything. So, that's this concentration 506 00:34:13 --> 00:34:19 model. It's the one entirely analogous 507 00:34:15 --> 00:34:21 to the temperature. And the physical setup is the 508 00:34:18 --> 00:34:24 same. This one is entirely different. 509 00:34:21 --> 00:34:27 Mixing in this form of this problem has really nothing to do 510 00:34:24 --> 00:34:30 with this model whatsoever. But, nor does that 511 00:34:27 --> 00:34:33 concentration had anything to do with this concentration, 512 00:34:31 --> 00:34:37 which refers to the result of the mixing in the tank. 513 00:34:36 --> 00:34:42 But, what happens is the differential equation is the 514 00:34:40 --> 00:34:46 same. The language of input and 515 00:34:43 --> 00:34:49 response that we talked about is also available here. 516 00:34:47 --> 00:34:53 So, everything is the same. And, the most interesting thing 517 00:34:52 --> 00:34:58 is that it shows that the analog of the conductivity, 518 00:34:57 --> 00:35:03 the k, the analog of conductivity and diffusivity is 519 00:35:01 --> 00:35:07 this quantity. I should not be considering r 520 00:35:06 --> 00:35:12 and v by themselves. I should be considering as the 521 00:35:11 --> 00:35:17 basic quantity, the ratio of those two. 522 00:35:15 --> 00:35:21 Now, why is that, is the basic parameter. 523 00:35:19 --> 00:35:25 What is this? Well, r is the rate of outflow, 524 00:35:23 --> 00:35:29 and the rate of inflow, what's r over v? 525 00:35:27 --> 00:35:33 r over v is the fractional rate of outflow. 526 00:35:31 --> 00:35:37 In other words, if r over v is one tenth, 527 00:35:35 --> 00:35:41 it means that 1/10 of the tank will be emptied in a minute, 528 00:35:40 --> 00:35:46 say. In other words, 529 00:35:44 --> 00:35:50 we lumped these two constants into a single k, 530 00:35:49 --> 00:35:55 and at the same time have simplified the units. 531 00:35:53 --> 00:35:59 What are the units? This is volume per minute. 532 00:35:58 --> 00:36:04 This is volume. So, it's simply reciprocal 533 00:36:02 --> 00:36:08 minutes, reciprocal time, which was the same units of 534 00:36:07 --> 00:36:13 that diffusivity and conductivity had, 535 00:36:11 --> 00:36:17 reciprocal time. The space variables have 536 00:36:16 --> 00:36:22 entirely disappeared. So, it that way, 537 00:36:19 --> 00:36:25 it's simplified. It's simplified conceptually, 538 00:36:22 --> 00:36:28 and now, you can answer the same type of questions we asked 539 00:36:27 --> 00:36:33 before about this. I think it would be better for 540 00:36:32 --> 00:36:38 us to move on, though. 541 00:36:33 --> 00:36:39 Well, just an example, one really simple thing, 542 00:36:37 --> 00:36:43 so, suppose since we spent so much time worrying about what 543 00:36:41 --> 00:36:47 was happening with sinusoid inputs, I mean, 544 00:36:45 --> 00:36:51 when could Ce be sinusoidal, for example? 545 00:36:48 --> 00:36:54 Well, roughly sinusoidal if, for example, 546 00:36:51 --> 00:36:57 some factory were polluting. If this were a lake, 547 00:36:55 --> 00:37:01 and some factory were polluting it, but in the beginning, 548 00:36:59 --> 00:37:05 at the beginning of the day, they produced a lot of the 549 00:37:03 --> 00:37:09 pollutant, and by the end of the day when it wound down, 550 00:37:07 --> 00:37:13 it might well happen that the concentration of pollutants in 551 00:37:12 --> 00:37:18 the incoming stream would vary sinusoidally with a 24 hour 552 00:37:16 --> 00:37:22 cycle. And then, we would be asking, 553 00:37:22 --> 00:37:28 so, suppose this varies sinusoidally. 554 00:37:26 --> 00:37:32 In other words, it's like cosine omega t. 555 00:37:31 --> 00:37:37 I'm asking, how closely does 556 00:37:36 --> 00:37:42 the concentration in the tank follow C sub e? 557 00:37:44 --> 00:37:50 Now, what would that depend upon? 558 00:37:48 --> 00:37:54 Think about it. Well, the answer, 559 00:37:52 --> 00:37:58 suppose k is large. Closely, let's just analyze one 560 00:37:58 --> 00:38:04 case, if k is big. Now, what would make k big? 561 00:38:03 --> 00:38:09 We know that from the temperature thing. 562 00:38:06 --> 00:38:12 If the conductivity is high, then the inner temperature will 563 00:38:10 --> 00:38:16 follow the outer temperature closely, and the same thing with 564 00:38:15 --> 00:38:21 the diffusion model. But we, of course, 565 00:38:17 --> 00:38:23 therefore, since the equation is the same, we must get the 566 00:38:21 --> 00:38:27 same result here. Now, what would make k big? 567 00:38:25 --> 00:38:31 If r is big, if the flow rate is very fast, 568 00:38:28 --> 00:38:34 we will expect the concentration of the inside of 569 00:38:31 --> 00:38:37 that tank to match fairly closely the concentration of the 570 00:38:35 --> 00:38:41 pollutant, of the incoming salt solution, or, 571 00:38:38 --> 00:38:44 if the tank is very small. For fixed flow rates, 572 00:38:43 --> 00:38:49 if the tank is very small, well, then it gets emptied 573 00:38:47 --> 00:38:53 quickly. So, both of these are, 574 00:38:49 --> 00:38:55 I think, intuitive results. And, of course, 575 00:38:52 --> 00:38:58 as before, we got them from that, by trying to analyze the 576 00:38:56 --> 00:39:02 final form of the solution. In other words, 577 00:38:59 --> 00:39:05 we got them by looking at that form of the solution up there, 578 00:39:03 --> 00:39:09 and seeing if k is big. As k increases, 579 00:39:07 --> 00:39:13 what happens to the amplitude, and what happens to the phase 580 00:39:12 --> 00:39:18 lag? But, that summarizes the two. 581 00:39:14 --> 00:39:20 So, this means, closely means, 582 00:39:17 --> 00:39:23 that the phase lag is, big or small? 583 00:39:20 --> 00:39:26 The lag is small. And, the amplitude is, 584 00:39:23 --> 00:39:29 well, the amplitude, the biggest the amplitude could 585 00:39:27 --> 00:39:33 ever be is one because that's the amplitude of this. 586 00:39:33 --> 00:39:39 So, the amplitude is near one, one because that's the 587 00:39:38 --> 00:39:44 amplitude of the incoming signal, input, 588 00:39:41 --> 00:39:47 whatever you want to call it. Okay, now, I'd like to spend 589 00:39:47 --> 00:39:53 the rest of the time talking about the failures of number 590 00:39:52 --> 00:39:58 one, and when you have to use number two, and when even number 591 00:39:58 --> 00:40:04 two is no good. So, let me end first-order 592 00:40:03 --> 00:40:09 equations by putting my worst foot forward. 593 00:40:08 --> 00:40:14 Well, I'm just trying to avoid disappointment at 594 00:40:13 --> 00:40:19 misapprehensions from you. I'll watch you leave this room 595 00:40:19 --> 00:40:25 and say, well, he said that, 596 00:40:22 --> 00:40:28 okay. So, the first one you're going 597 00:40:26 --> 00:40:32 to encounter very shortly where one is not satisfied, 598 00:40:32 --> 00:40:38 but two is, so on some examples where you need two, 599 00:40:38 --> 00:40:44 well, it's going to happen right here. 600 00:40:44 --> 00:40:50 Somebody, sooner or later, it's going to draw on that 601 00:40:47 --> 00:40:53 loathsome orange chalk, which is unerasable, 602 00:40:50 --> 00:40:56 something which looks like that. 603 00:40:52 --> 00:40:58 Remember, you saw it here first. 604 00:40:54 --> 00:41:00 r, yeah, we had that. Okay, see, I had it in high 605 00:40:58 --> 00:41:04 school too. That's the capacitance. 606 00:41:00 --> 00:41:06 This is the resistance. That's the electromotive force: 607 00:41:04 --> 00:41:10 battery, or a source of alternating current, 608 00:41:07 --> 00:41:13 something like that. Now, of course, 609 00:41:11 --> 00:41:17 what you're interested in is how the current flows in the 610 00:41:15 --> 00:41:21 circuit. Since current across the 611 00:41:18 --> 00:41:24 capacitance doesn't make sense, you have to talk about the 612 00:41:23 --> 00:41:29 charge on the capacitance. So, q, it's customary in a 613 00:41:27 --> 00:41:33 circle this simple to use as the variable not current, 614 00:41:32 --> 00:41:38 but the charge on the capacitance. 615 00:41:36 --> 00:41:42 And then Kirchhoff's, you are also supposed to know 616 00:41:40 --> 00:41:46 that the derivative, that the time derivative of q 617 00:41:45 --> 00:41:51 is what's called the current in the circuit. 618 00:41:49 --> 00:41:55 That sort of intuitive. But, i in a physics class, 619 00:41:53 --> 00:41:59 j in an electrical engineering class, and why, 620 00:41:57 --> 00:42:03 not the letter Y, but why is that? 621 00:42:02 --> 00:42:08 That's because of electrical engineers use lots of lots of 622 00:42:06 --> 00:42:12 complex numbers and therefore, you have to call current j, 623 00:42:11 --> 00:42:17 I guess. I think they do j in physics, 624 00:42:15 --> 00:42:21 too, now. No, no they don't. 625 00:42:17 --> 00:42:23 I don't know. So, i is ambiguous if you are 626 00:42:21 --> 00:42:27 in that particular subject. And it's customary to use, 627 00:42:25 --> 00:42:31 I don't know. Now it's completely safe. 628 00:42:29 --> 00:42:35 Okay, where are we? Well, the law is, 629 00:42:32 --> 00:42:38 the basic differential equation is Kirchhoff's voltage law, 630 00:42:36 --> 00:42:42 but the sum of the voltage drops across these three has to 631 00:42:40 --> 00:42:46 be zero. So, it's R times i, 632 00:42:42 --> 00:42:48 which is dq / dt. That's Ohm's law. 633 00:42:44 --> 00:42:50 That's the voltage drop across resistance. 634 00:42:47 --> 00:42:53 The voltage drop across the capacitance is Coulomb's law, 635 00:42:51 --> 00:42:57 one form of Coulomb's law. It's q divided by C. 636 00:42:55 --> 00:43:01 And, that has to be the voltage drop. 637 00:42:57 --> 00:43:03 And then, there is some sign convention. 638 00:43:02 --> 00:43:08 So, this is either plus or minus, depending on your sign 639 00:43:06 --> 00:43:12 conventions, but it's E of t. Now, if I put that in standard 640 00:43:11 --> 00:43:17 form, in standard form I probably should say q prime plus 641 00:43:15 --> 00:43:21 q over RC equals, well, I suppose, 642 00:43:18 --> 00:43:24 E over R. And, this is what would appear 643 00:43:23 --> 00:43:29 in the equation. But, it's not the natural 644 00:43:26 --> 00:43:32 thing. The k is one over RC. 645 00:43:30 --> 00:43:36 And, that's the reciprocal. 646 00:43:32 --> 00:43:38 The RC constant is what everybody knows is important 647 00:43:35 --> 00:43:41 when you talk about a little circuit of that form. 648 00:43:39 --> 00:43:45 On the other hand, the right-hand side, 649 00:43:41 --> 00:43:47 it's quite unnatural to try to stick in the right-hand side 650 00:43:46 --> 00:43:52 that same RC. Call this EC over RC. 651 00:43:48 --> 00:43:54 People don't do that, and therefore, 652 00:43:50 --> 00:43:56 it doesn't really fall into the paradigm of that first equation. 653 00:43:55 --> 00:44:01 It's the second equation that really falls into the category. 654 00:43:59 --> 00:44:05 Another simple example of this is chained to k, 655 00:44:02 --> 00:44:08 radioactively changed to k. Well, let's say the radioactive 656 00:44:08 --> 00:44:14 substance, A, decays into, 657 00:44:10 --> 00:44:16 let's say, one atom of this produces one atom of that for 658 00:44:14 --> 00:44:20 simplicity. So, it decays into B, 659 00:44:17 --> 00:44:23 which then still is radioactive and decays further. 660 00:44:21 --> 00:44:27 Okay, what's the differential equation, which is going to be, 661 00:44:26 --> 00:44:32 it's going to govern this situation? 662 00:44:29 --> 00:44:35 What I want to know is how much B there is at any given time. 663 00:44:35 --> 00:44:41 So, I want a differential equation for the quantity of the 664 00:44:39 --> 00:44:45 radioactive product at any given time. 665 00:44:41 --> 00:44:47 Well, what's it going to look like? 666 00:44:44 --> 00:44:50 Well, it's the amount coming in minus the amount going out, 667 00:44:48 --> 00:44:54 so to speak. The rate of inflow minus the 668 00:44:51 --> 00:44:57 rate of outflow, except it's not the same type 669 00:44:55 --> 00:45:01 of physical flow we had before. How fast is it coming in? 670 00:44:59 --> 00:45:05 Well, A is decaying at a certain rate, 671 00:45:01 --> 00:45:07 and so the rate at which A decays is by the basic 672 00:45:05 --> 00:45:11 radioactive law. It's k1, it's constant, 673 00:45:09 --> 00:45:15 decay constant, times the amount of A present. 674 00:45:12 --> 00:45:18 If I used the differential 675 00:45:15 --> 00:45:21 equation with A here, I'd have to put a negative sign 676 00:45:18 --> 00:45:24 because it's the rate at which that stuff is leaving A. 677 00:45:22 --> 00:45:28 But, I'm interested in the rate at which it's coming in to B. 678 00:45:26 --> 00:45:32 So, it has a positive sign. And then, the rate at which B 679 00:45:30 --> 00:45:36 is decaying, and therefore the quantity of good B is gone, 680 00:45:34 --> 00:45:40 -- -- that will have some other 681 00:45:38 --> 00:45:44 constant, B. So, that will be the equation, 682 00:45:41 --> 00:45:47 and to avoid having two dependent variables in there, 683 00:45:46 --> 00:45:52 we know how A is decaying. So, it's k1, 684 00:45:49 --> 00:45:55 some constant times A, sorry, A will be e to the 685 00:45:53 --> 00:45:59 negative, you know, the decay law, 686 00:45:56 --> 00:46:02 so, times the initial amount that was there times e to the 687 00:46:01 --> 00:46:07 negative k1 t. That's how much A there is at 688 00:46:07 --> 00:46:13 any given time. It's decaying by the 689 00:46:10 --> 00:46:16 radioactive decay law, minus k2 B. 690 00:46:14 --> 00:46:20 Okay, so how does the differential equation look like? 691 00:46:18 --> 00:46:24 It looks like B prime plus k2 B equals an exponential, 692 00:46:23 --> 00:46:29 k1 A zero e to the negative k1 t. 693 00:46:28 --> 00:46:34 But, there's no reason to 694 00:46:31 --> 00:46:37 expect that that constant really has anything to do with k2. 695 00:46:36 --> 00:46:42 It's unnatural to put it in that form, which is the correct 696 00:46:41 --> 00:46:47 one. Now, in the last two minutes, 697 00:46:49 --> 00:46:55 I wish to alienate half the class by pointing out that if k 698 00:47:02 --> 00:47:08 is less than zero, none of the terminology of 699 00:47:13 --> 00:47:19 transient, steady-state input response applies. 700 00:47:25 --> 00:47:31 The technique of solving the equation is identical. 701 00:47:28 --> 00:47:34 But, you cannot interpret. So, the technique is the same, 702 00:47:33 --> 00:47:39 and therefore it's worth learning. 703 00:47:37 --> 00:47:43 The technique is the same. In other words, 704 00:47:41 --> 00:47:47 the solution will be still e to the negative kt integral q of t 705 00:47:48 --> 00:47:54 e to the kt dt plus 706 00:47:54 --> 00:48:00 a constant times e to the k, oh, this is terrible, 707 00:48:00 --> 00:48:06 no. dy / dt, let's give an example. 708 00:48:04 --> 00:48:10 The equation I'm going to look at is something that looks like 709 00:48:10 --> 00:48:16 this: y equals q of t, let's say, okay, 710 00:48:13 --> 00:48:19 a constant, but the constant a is positive. 711 00:48:17 --> 00:48:23 So, the constant here is negative. 712 00:48:19 --> 00:48:25 Then, when I solve, my k, in other words, 713 00:48:23 --> 00:48:29 is now properly written as negative a. 714 00:48:26 --> 00:48:32 And therefore, this formula should now become 715 00:48:30 --> 00:48:36 not this, but the negative k is a t. 716 00:48:36 --> 00:48:42 And, here it's negative a t. And, here it is positive a t. 717 00:48:41 --> 00:48:47 Now, why is it, if this is going to be the 718 00:48:44 --> 00:48:50 solution, why are all those things totally irrelevant? 719 00:48:49 --> 00:48:55 This is not a transient any longer because if a is positive, 720 00:48:54 --> 00:49:00 this goes to infinity. Or, if I go to minus infinity, 721 00:48:59 --> 00:49:05 then C is negative. So, it's not transient. 722 00:49:03 --> 00:49:09 It's not going to zero, and it depends heavily on the 723 00:49:07 --> 00:49:13 initial conditions. That means that of these two 724 00:49:10 --> 00:49:16 functions, this is the important guy. 725 00:49:12 --> 00:49:18 This is just fixed, some fixed function. 726 00:49:15 --> 00:49:21 Everything, in other words, is going to depend upon the 727 00:49:18 --> 00:49:24 initial conditions, whereas in the other cases we 728 00:49:21 --> 00:49:27 have been studying, the initial conditions after a 729 00:49:25 --> 00:49:31 while don't matter anymore. Now, why did I say I would 730 00:49:30 --> 00:49:36 alienate half of you? Well, because in what subjects 731 00:49:35 --> 00:49:41 will a be positive? In what subjects will k be 732 00:49:39 --> 00:49:45 negative? k is typically negative in 733 00:49:43 --> 00:49:49 biology, economics, Sloan. 734 00:49:45 --> 00:49:51 In other words, the simple thing is think of it 735 00:49:50 --> 00:49:56 in biology. What's the simplest equation 736 00:49:53 --> 00:49:59 for population growth? Well, it is dP / dt equals 737 00:49:58 --> 00:50:04 some, if the population is growing, a times P, 738 00:50:02 --> 00:50:08 and a is a positive number. That means P prime minus a P is 739 00:50:09 --> 00:50:15 zero. So, the thing I want to leave 740 00:50:13 --> 00:50:19 you with is this. If life is involved, 741 00:50:16 --> 00:50:22 k is likely to be negative. k is positive when inanimate 742 00:50:22 --> 00:50:28 things are involved; I won't say dead, 743 00:50:25 --> 00:50:31 inanimate.