1 00:00:07,155 --> 00:00:10,740 PROFESSOR: All right, I would like to then get back to a 2 00:00:10,740 --> 00:00:15,680 discussion of some of the basic relations that we have 3 00:00:15,680 --> 00:00:18,390 been discussing. 4 00:00:18,390 --> 00:00:22,640 We didn't get terribly far, but I'd like to start with the 5 00:00:22,640 --> 00:00:28,390 Cartesian coordinate system that we set up. 6 00:00:28,390 --> 00:00:31,410 Rather than using x, y, and z, I'm labeling the 7 00:00:31,410 --> 00:00:33,990 axes x1, x2, and x3. 8 00:00:33,990 --> 00:00:39,150 And we'll see that the subscripts play a very useful 9 00:00:39,150 --> 00:00:43,860 role in the formalism we're about to develop. 10 00:00:43,860 --> 00:00:46,880 Now, the first thing we might want to specify in this 11 00:00:46,880 --> 00:00:53,160 coordinate is the orientation of a vector and its 12 00:00:53,160 --> 00:00:54,390 components. 13 00:00:54,390 --> 00:01:01,130 So let's suppose that this is some vector P. And what I will 14 00:01:01,130 --> 00:01:06,850 do to define its orientation is to use the three angles 15 00:01:06,850 --> 00:01:12,350 that the vector makes, or the direction makes with respect 16 00:01:12,350 --> 00:01:14,590 to x1, x2, x3. 17 00:01:14,590 --> 00:01:18,580 And we could define these angles as theta1, that's the 18 00:01:18,580 --> 00:01:24,950 angle between the direction and x1, theta2, the angle 19 00:01:24,950 --> 00:01:29,590 between our direction or our vector and x2, and finally, 20 00:01:29,590 --> 00:01:31,880 not surprisingly, I'll call this one theta3. 21 00:01:34,520 --> 00:01:42,040 So the three components of the vector could be written as P1, 22 00:01:42,040 --> 00:01:47,370 the component along x1 is going to be the magnitude of P 23 00:01:47,370 --> 00:01:52,000 times the cosine of theta1. 24 00:01:52,000 --> 00:01:57,080 The x2 component of P would be the magnitude of P times the 25 00:01:57,080 --> 00:02:00,840 cosine of theta2. 26 00:02:00,840 --> 00:02:04,830 And P3, the third component, would be the magnitude of P 27 00:02:04,830 --> 00:02:06,680 times the cosine of theta3. 28 00:02:11,360 --> 00:02:18,450 Now, we will have so many relations that involve the 29 00:02:18,450 --> 00:02:22,720 cosine of the angle between a direction and one of our 30 00:02:22,720 --> 00:02:27,770 reference axes that it is convenient to define a special 31 00:02:27,770 --> 00:02:30,820 term for the cosines of these angles. 32 00:02:30,820 --> 00:02:34,450 So I'll define this as magnitude of P times the 33 00:02:34,450 --> 00:02:40,400 quantity l1, magnitude of P times l2, magnitude of P times 34 00:02:40,400 --> 00:02:42,960 l3, which is a lot easier to write. 35 00:02:42,960 --> 00:02:44,900 And we will define these things as 36 00:02:44,900 --> 00:02:46,150 the direction cosines. 37 00:02:54,300 --> 00:03:00,060 With these equations it's easy to attach some meaning to the 38 00:03:00,060 --> 00:03:01,740 direction cosines. 39 00:03:01,740 --> 00:03:09,010 Suppose we had a vector of magnitude 1, something that we 40 00:03:09,010 --> 00:03:10,790 will refer to as a unit vector. 41 00:03:13,860 --> 00:03:20,890 And if we put in magnitude of P equal to 1, it follows that 42 00:03:20,890 --> 00:03:35,880 l1m l2m l3 are simply the components of a unit vector in 43 00:03:35,880 --> 00:03:52,460 a particular direction along, obviously, x1, x2, and x3, 44 00:03:52,460 --> 00:03:53,710 respectively. 45 00:03:55,600 --> 00:04:01,200 Trivial piece of algebra, but it attaches a physical and 46 00:04:01,200 --> 00:04:03,870 geometric significance to the direction cosines. 47 00:04:09,840 --> 00:04:14,490 Now, the vector is something that could represent a 48 00:04:14,490 --> 00:04:15,820 physical quantity. 49 00:04:15,820 --> 00:04:19,850 In any case, it is something that is absolute. 50 00:04:19,850 --> 00:04:25,350 And it sits embedded majestically, relative to some 51 00:04:25,350 --> 00:04:27,710 absolute coordinate system. 52 00:04:27,710 --> 00:04:32,030 The magnitudes of the components P1, P2, and P3 will 53 00:04:32,030 --> 00:04:35,980 change their values if we would decide to change the 54 00:04:35,980 --> 00:04:39,690 coordinate system that we're using as our reference system. 55 00:04:39,690 --> 00:04:42,970 So the next question we might ask is, suppose we change the 56 00:04:42,970 --> 00:04:53,345 coordinate system to some new values, x1 prime, x2 prime, 57 00:04:53,345 --> 00:04:55,770 and x3 prime? 58 00:04:55,770 --> 00:05:00,220 And I'll illustrate my point with just a two dimensional 59 00:05:00,220 --> 00:05:01,430 analog of this. 60 00:05:01,430 --> 00:05:03,890 This is x1, and this is x2. 61 00:05:03,890 --> 00:05:06,530 And this is my vector P. 62 00:05:06,530 --> 00:05:12,950 And I change x1 to some new value, x1 prime, and change x2 63 00:05:12,950 --> 00:05:16,170 to some new orientation, x2 prime. 64 00:05:16,170 --> 00:05:21,620 Then clearly the component of P on x1 has changed its 65 00:05:21,620 --> 00:05:27,070 numerical value if I refer it to x1 prime instead. 66 00:05:27,070 --> 00:05:32,050 And similarly, this value would be the component P2. 67 00:05:32,050 --> 00:05:36,440 If I change the direction of x2 and draw a perpendicular to 68 00:05:36,440 --> 00:05:39,100 x2, this would be P2 prime. 69 00:05:39,100 --> 00:05:41,670 So if I change coordinate system along the fashion I 70 00:05:41,670 --> 00:05:45,380 suggested, the three components of a vector, P1, 71 00:05:45,380 --> 00:05:49,850 P2, P3, are going to change to some new values, P1 prime, P2 72 00:05:49,850 --> 00:05:51,100 prime, P3 prime. 73 00:05:57,060 --> 00:05:57,280 OK. 74 00:05:57,280 --> 00:06:01,860 So the question I'd like to address next is given the 75 00:06:01,860 --> 00:06:07,240 change of coordinate system, and given the three components 76 00:06:07,240 --> 00:06:11,750 of P in the original coordinate system, how do I 77 00:06:11,750 --> 00:06:15,670 compute the values of the new components P1 prime, 78 00:06:15,670 --> 00:06:16,920 P2 prime, P3 prime? 79 00:06:20,660 --> 00:06:29,820 I'll say it in words, and then we'll define a mechanism for 80 00:06:29,820 --> 00:06:33,350 specifying the change in coordinate system. 81 00:06:33,350 --> 00:06:35,780 What I'll say is-- and this was apparent in the sketch 82 00:06:35,780 --> 00:06:41,710 that I just erased-- 83 00:06:41,710 --> 00:06:45,500 the new component of the vector P1 prime is simply 84 00:06:45,500 --> 00:06:55,300 going to be the sum of the components of P1, P2, and P3 85 00:06:55,300 --> 00:06:57,530 along the new x prime direction. 86 00:06:57,530 --> 00:07:01,080 So I'm saying that this is going to be the sum of the 87 00:07:01,080 --> 00:07:13,200 component of P1 along x1 plus the component of P2 along x1 88 00:07:13,200 --> 00:07:28,950 prime and the component of P3 along x1 prime. 89 00:07:28,950 --> 00:07:31,980 So in short, I'm doing nothing more complicated than saying, 90 00:07:31,980 --> 00:07:36,020 I can get the values of the new components if I take the 91 00:07:36,020 --> 00:07:41,380 vector P, split it up into its three parts, and then find the 92 00:07:41,380 --> 00:07:45,960 component of each of these three parts along the x1 prime 93 00:07:45,960 --> 00:07:50,340 access, then do the same thing for the x2 prime axis, and 94 00:07:50,340 --> 00:07:52,362 then the same thing for x3 prime. 95 00:07:58,490 --> 00:08:05,490 So I'm going to need, now, a notation for a change in a 96 00:08:05,490 --> 00:08:08,060 three-dimensional Cartesian coordinate system. 97 00:08:12,630 --> 00:08:19,670 So here is x1, here is x2, and here is x3. 98 00:08:19,670 --> 00:08:21,180 And I will change them. 99 00:08:21,180 --> 00:08:22,580 And again, I'm always keeping the 100 00:08:22,580 --> 00:08:23,830 coordinate system Cartesian. 101 00:08:27,230 --> 00:08:32,250 So here's an x1 prime, here's an x2 prime, and then x3 prime 102 00:08:32,250 --> 00:08:34,520 will point out in some direction like this. 103 00:08:34,520 --> 00:08:35,978 I don't want a prime on that. 104 00:08:38,900 --> 00:08:45,110 So I'm going to say now that the component of P along the 105 00:08:45,110 --> 00:08:51,500 new x1 prime is going to be the magnitude of P1 times the 106 00:08:51,500 --> 00:09:05,500 cosine of the angle between x1 and x1 prime, plus P2 times 107 00:09:05,500 --> 00:09:12,120 the cosine of the angle between x1 and-- 108 00:09:12,120 --> 00:09:13,030 am I doing this right? 109 00:09:13,030 --> 00:09:17,280 P1 onto x1 prime. 110 00:09:17,280 --> 00:09:22,200 And I want P2 onto x1 prime. 111 00:09:22,200 --> 00:09:27,740 So this is going to be the angle between x2 and x1 prime 112 00:09:27,740 --> 00:09:31,520 plus P3 times the cosine of the angle 113 00:09:31,520 --> 00:09:34,180 between x3 and x1 prime. 114 00:09:41,230 --> 00:09:46,520 Well, we used C's or l earlier on to 115 00:09:46,520 --> 00:09:49,490 represent a direction cosine. 116 00:09:49,490 --> 00:10:14,450 Let me define Cij as the cosine of the angle between x1 117 00:10:14,450 --> 00:10:19,670 prime, xi prime, and x sub j. 118 00:10:23,390 --> 00:10:31,490 So that means I can write this expression here in this nice 119 00:10:31,490 --> 00:10:32,430 compact form. 120 00:10:32,430 --> 00:10:34,890 With our definition of direction cosines, I can say 121 00:10:34,890 --> 00:10:43,472 that P1 prime is going to be equal to C1 1 times P1 plus C1 122 00:10:43,472 --> 00:10:53,490 2 times P2 plus C1 3 times P3. 123 00:10:53,490 --> 00:10:56,610 Can write that P2 prime in the same way. 124 00:10:56,610 --> 00:11:04,340 It's going to be the cosine of the angle between P2 prime and 125 00:11:04,340 --> 00:11:10,570 P1 and x1, plus the cosine of the angle between x2 prime and 126 00:11:10,570 --> 00:11:16,680 x2 times P2 plus C2 3 which is the cosine of the angle 127 00:11:16,680 --> 00:11:21,710 between x2 prime and x3, times the component P3. 128 00:11:21,710 --> 00:11:26,420 And in a very similar fashion, P3 prime will be C3 1 P1 plus 129 00:11:26,420 --> 00:11:31,215 C3 2 times P2 plus C3 3 times P3. 130 00:11:35,030 --> 00:11:37,545 So here is the way a vector will transform. 131 00:11:43,950 --> 00:11:48,190 And we can write this compactly in matrix form. 132 00:11:48,190 --> 00:11:56,580 We can say that P sub i prime, where this is a column matrix, 133 00:11:56,580 --> 00:12:03,920 one by three, is going to be equal to Cij, a three by three 134 00:12:03,920 --> 00:12:06,780 matrix times the original components of the 135 00:12:06,780 --> 00:12:08,030 vector P sub j. 136 00:12:12,450 --> 00:12:15,570 And just to cement the notation that we're using, if 137 00:12:15,570 --> 00:12:30,480 I put my old axes up here, x1, x2, x3, and put the new axes, 138 00:12:30,480 --> 00:12:39,100 x1 prime, x2 prime, x3 prime, down this way, then the cosine 139 00:12:39,100 --> 00:12:47,010 of the angle between the quantities that are in this 140 00:12:47,010 --> 00:12:53,050 column and the quantities that are in this row would be C1 1, 141 00:12:53,050 --> 00:13:03,820 C1 2, C1 3, C2 1, C2 2, C2 3, C3 1, C3 2, C3 3. 142 00:13:11,730 --> 00:13:13,540 Nothing fancy except the notation. 143 00:13:13,540 --> 00:13:17,280 It's the description of some very simple geometry. 144 00:13:19,920 --> 00:13:24,590 This array, Cij, is something that I will refer to as a 145 00:13:24,590 --> 00:13:26,190 direction cosine scheme. 146 00:13:55,440 --> 00:14:01,240 Let me pause here and see if that's all sunk in, whether 147 00:14:01,240 --> 00:14:02,680 you have any questions on this. 148 00:14:08,170 --> 00:14:13,270 One of the nasty properties of what we're going to be doing 149 00:14:13,270 --> 00:14:17,740 for the next month or so is that the notions are really 150 00:14:17,740 --> 00:14:21,650 very, very simple, but the notation is horribly 151 00:14:21,650 --> 00:14:23,340 cumbersome and complex. 152 00:14:23,340 --> 00:14:26,820 So it takes a bit of getting used to in application to 153 00:14:26,820 --> 00:14:30,640 actual real cases before you feel fully at home with it. 154 00:14:35,470 --> 00:14:36,150 OK. 155 00:14:36,150 --> 00:14:37,850 Just a matter of definition so far. 156 00:14:58,930 --> 00:15:03,160 Let me note that this direction cosine array is 157 00:15:03,160 --> 00:15:08,510 going to be useful for defining how a vector changes 158 00:15:08,510 --> 00:15:11,800 as we go from the original coordinate system to a new 159 00:15:11,800 --> 00:15:13,120 coordinate system. 160 00:15:13,120 --> 00:15:19,040 But this direction cosine array will also tell us how 161 00:15:19,040 --> 00:15:25,630 the axes in one coordinate system are related to the axes 162 00:15:25,630 --> 00:15:28,210 in the new coordinate system. 163 00:15:28,210 --> 00:15:32,140 It follows from the fact that the axes themselves can be 164 00:15:32,140 --> 00:15:35,250 regarded as unit vectors. 165 00:15:35,250 --> 00:15:42,350 And we said that the components of a unit vector 166 00:15:42,350 --> 00:15:47,590 are the direction cosines of that vector, relative to a 167 00:15:47,590 --> 00:15:49,410 coordinate system. 168 00:15:49,410 --> 00:15:56,720 So let's ask, what are the new components of x1 in terms of 169 00:15:56,720 --> 00:16:01,640 the original axes unprimed. 170 00:16:01,640 --> 00:16:08,780 Well, x1 prime is going to be the unit vector x1 times the 171 00:16:08,780 --> 00:16:16,280 cosine of the angle between x1 and x1 prime, plus the unit 172 00:16:16,280 --> 00:16:19,940 vector along x2 prime times the cosine of the angle 173 00:16:19,940 --> 00:16:22,130 between x1 prime, and x2. 174 00:16:22,130 --> 00:16:23,830 And that's C1 2. 175 00:16:23,830 --> 00:16:29,760 Plus x3 regarded as a unit vector times the cosine of the 176 00:16:29,760 --> 00:16:32,490 angle between x1 prime and x3. 177 00:16:32,490 --> 00:16:35,580 So we can actually write an equation for unit vectors 178 00:16:35,580 --> 00:16:37,480 along each of our new axes. 179 00:16:37,480 --> 00:16:45,120 And they will go as C1 1 times x1, C1 2 times x2, C1 3 times 180 00:16:45,120 --> 00:16:54,200 x3, plus C2 2 times x2 plus C2 3 times x3 times x3 prime. 181 00:16:54,200 --> 00:17:01,910 And x3 primal will be C3 1 x1 plus C3 2 x2 plus C3 3 x3. 182 00:17:01,910 --> 00:17:08,200 So this, then, is an equation between the unit vectors along 183 00:17:08,200 --> 00:17:11,660 the three reference axes in the new coordinate system 184 00:17:11,660 --> 00:17:15,329 relative to those in the original coordinate system. 185 00:17:15,329 --> 00:17:17,829 And the direction cosine scheme does the job. 186 00:17:37,290 --> 00:17:38,540 OK. 187 00:17:41,860 --> 00:17:48,750 We could, using the same argument, give the array that 188 00:17:48,750 --> 00:17:51,370 specifies the reverse transformation. 189 00:17:51,370 --> 00:17:54,120 If we would change our mind, for example, and say we don't 190 00:17:54,120 --> 00:17:57,460 like what we've done, let's write the original coordinate 191 00:17:57,460 --> 00:18:02,050 system x1, x2, and x3 in terms of the unit vectors 192 00:18:02,050 --> 00:18:06,030 along the new axes. 193 00:18:06,030 --> 00:18:07,970 And we can use exactly the same array. 194 00:18:07,970 --> 00:18:13,430 We can say that the original x1, in terms of the three new 195 00:18:13,430 --> 00:18:18,520 axes, x1 prime, x2 prime, and x3 prime, is going to involve 196 00:18:18,520 --> 00:18:22,330 the cosine of the angle between x1 prime and x1, and 197 00:18:22,330 --> 00:18:28,750 that is C1 1, plus the cosine of the angle between x2 prime 198 00:18:28,750 --> 00:18:37,250 and x1, and that's C2 1, plus the cosine of the angle 199 00:18:37,250 --> 00:18:41,260 between x3 prime and x1, and that C3 1. 200 00:18:41,260 --> 00:18:46,310 If we continue on this, if you have the idea, the angle x2 is 201 00:18:46,310 --> 00:18:50,390 going to be given in terms of x1 prime, x2 prime, and x3 202 00:18:50,390 --> 00:18:55,950 prime, as the cosine of the angle between x2 and x1 prime, 203 00:18:55,950 --> 00:18:57,205 and that is C1 2. 204 00:19:00,180 --> 00:19:04,060 Here we want the cosine of the angle between x2 and x2 prime, 205 00:19:04,060 --> 00:19:13,610 and here the cosine of the angle between x3 prime and x2. 206 00:19:13,610 --> 00:19:16,460 And you can see the way this is playing out. 207 00:19:16,460 --> 00:19:22,900 C1 3 times x1 plus C2 3, x2 prime plus 208 00:19:22,900 --> 00:19:25,720 C3 3 times x3 prime. 209 00:19:25,720 --> 00:19:28,860 So there's the reverse transformation, using the same 210 00:19:28,860 --> 00:19:31,480 array of coefficients as we did the first time. 211 00:19:31,480 --> 00:19:34,480 So it turns out if we write this symbolically in a compact 212 00:19:34,480 --> 00:19:38,640 form, xi prime is given by Cijx sub j. 213 00:19:45,730 --> 00:19:49,360 And the reverse transformation using the same direction 214 00:19:49,360 --> 00:20:05,190 cosine says that xi is going to be Cji times x sub j prime. 215 00:20:05,190 --> 00:20:08,290 In other words, the reverse transformation, let's write it 216 00:20:08,290 --> 00:20:12,900 as Cij minus 1, the inverse transformation, turns out to 217 00:20:12,900 --> 00:20:15,950 be simply Cji. 218 00:20:15,950 --> 00:20:21,100 And that, in matrix algebra, is written as the transpose of 219 00:20:21,100 --> 00:20:23,810 the original array of coefficients. 220 00:20:23,810 --> 00:20:28,730 And transpose is either given by a squiggle, a tilde on top 221 00:20:28,730 --> 00:20:30,120 of the matrix. 222 00:20:30,120 --> 00:20:35,950 Some people like to use a superscript T. But we'll use 223 00:20:35,950 --> 00:20:37,540 this particular notation. 224 00:20:37,540 --> 00:20:39,820 But you can see either notation used to 225 00:20:39,820 --> 00:20:41,210 indicate the transpose. 226 00:20:49,180 --> 00:20:55,170 The array Cij, which has this property, and it also has 227 00:20:55,170 --> 00:20:58,580 another property which I won't bother to prove, but the 228 00:20:58,580 --> 00:21:03,240 determinant of Cij is equal to 1. 229 00:21:03,240 --> 00:21:05,825 And this is something called a unitary matrix. 230 00:21:12,010 --> 00:21:15,110 Unitary matrix has the property that the inverse 231 00:21:15,110 --> 00:21:16,665 matrix is the transpose. 232 00:21:20,890 --> 00:21:25,280 We will very, very shortly start writing down numbers for 233 00:21:25,280 --> 00:21:27,080 some specific transformations. 234 00:21:27,080 --> 00:21:30,670 And then I think that will give us a little facility in 235 00:21:30,670 --> 00:21:31,920 doing these manipulations. 236 00:21:42,270 --> 00:21:44,670 Comments or questions? 237 00:21:44,670 --> 00:21:49,080 Is this old stuff or old stuff for which the notation is 238 00:21:49,080 --> 00:21:50,330 still confusing? 239 00:21:57,470 --> 00:21:58,410 All right. 240 00:21:58,410 --> 00:22:01,670 Let me point out something that is 241 00:22:01,670 --> 00:22:03,640 perhaps apparent to you. 242 00:22:03,640 --> 00:22:08,370 And that is that not all nine of these numbers are 243 00:22:08,370 --> 00:22:10,350 independent. 244 00:22:10,350 --> 00:22:13,670 There are relations between them. 245 00:22:13,670 --> 00:22:17,270 And let's point out some of these relations. 246 00:22:24,890 --> 00:22:38,570 C1 1, C1 2, C1 2 represent the components of a unit vector 247 00:22:38,570 --> 00:22:45,870 along x1 prime, in the original coordinate system of 248 00:22:45,870 --> 00:22:56,470 the elements in any row is equal to 1. 249 00:22:56,470 --> 00:23:03,010 Because these are the components of a unit vector. 250 00:23:03,010 --> 00:23:05,160 And the magnitude of a unit vector is 1. 251 00:23:08,490 --> 00:23:11,580 In the same way, if we look at any column of terms in this 252 00:23:11,580 --> 00:23:18,700 matrix, for example, C1 1, C2 1, C3 1, these are terms that 253 00:23:18,700 --> 00:23:23,100 represent the cosine of angles between x1 in the original 254 00:23:23,100 --> 00:23:27,250 coordinate system and x1 prime, x2 prime, x3 prime, our 255 00:23:27,250 --> 00:23:28,940 new coordinate system. 256 00:23:28,940 --> 00:23:32,520 So this gives us the magnitude of x1, but 257 00:23:32,520 --> 00:23:34,030 x1 is a unit vector. 258 00:23:34,030 --> 00:23:39,960 So it follows that the sum of the column C1 1 squared plus 259 00:23:39,960 --> 00:23:45,750 C2 1 squared plus C3 1 squared also has to be unity, because 260 00:23:45,750 --> 00:23:54,070 that gives us the magnitude of a unit vector along x1, So the 261 00:23:54,070 --> 00:24:01,190 sum of the squares of elements in any column of the direction 262 00:24:01,190 --> 00:24:10,420 cosine is unity. 263 00:24:21,560 --> 00:24:24,000 These expressions are useful. 264 00:24:24,000 --> 00:24:27,120 But they have one ambiguity. 265 00:24:27,120 --> 00:24:31,690 That is the cosine of an angle can be either positive or 266 00:24:31,690 --> 00:24:35,760 negative, depending on whether the angle is less than 90 267 00:24:35,760 --> 00:24:38,710 degrees or greater than 90 degrees. 268 00:24:38,710 --> 00:24:42,780 These relations involve the squares of direction cosines, 269 00:24:42,780 --> 00:24:47,110 and therefore we can't tell whether the direction cosine 270 00:24:47,110 --> 00:24:52,860 itself is positive or negative. 271 00:24:52,860 --> 00:24:56,400 So let me put down a limitation here. 272 00:24:56,400 --> 00:25:01,010 And that is we cannot tell the sign. 273 00:25:05,410 --> 00:25:08,890 Every time I point this out to people I wince inside. 274 00:25:08,890 --> 00:25:13,460 Because I once spent two weeks trying to debug a computer 275 00:25:13,460 --> 00:25:15,540 program, and it wasn't working. 276 00:25:15,540 --> 00:25:18,380 And it turns out the reason it wasn't working properly was 277 00:25:18,380 --> 00:25:23,010 that I didn't realize that you cannot tell the sign when all 278 00:25:23,010 --> 00:25:25,900 you know is the squares of the direction cosines. 279 00:25:25,900 --> 00:25:31,305 So I remember this as a rather pointed observation. 280 00:25:36,020 --> 00:25:40,440 Happily, there are other relations among this array of 281 00:25:40,440 --> 00:25:41,690 coefficients. 282 00:25:49,170 --> 00:25:56,610 This row of terms represents the components of x1 prime in 283 00:25:56,610 --> 00:26:00,760 the original coordinate system x1, x2, x3. 284 00:26:00,760 --> 00:26:08,030 This row immediately below it represents the components of a 285 00:26:08,030 --> 00:26:12,020 unit vector x2 prime relative to the original coordinate 286 00:26:12,020 --> 00:26:18,170 system x1, x2, x3. 287 00:26:18,170 --> 00:26:22,290 Our coordinate systems are Cartesian. 288 00:26:22,290 --> 00:26:27,960 Therefore, this unit vector has to be perpendicular to the 289 00:26:27,960 --> 00:26:30,580 unit vector along x2 prime. 290 00:26:30,580 --> 00:26:35,020 And that means their dot product has to be 0. 291 00:26:35,020 --> 00:26:38,220 So let me indicate that this way. 292 00:26:38,220 --> 00:26:42,090 The unit vector along x1 prime dotted with the unit vector 293 00:26:42,090 --> 00:26:45,420 along x2 prime has to be 0. 294 00:26:45,420 --> 00:26:55,710 And that dot product is going to be C1 1 times C2 1, that's 295 00:26:55,710 --> 00:27:02,150 the product of these two terms, plus C1 2 times C2 2 296 00:27:02,150 --> 00:27:05,840 plus C1 3 times C2 3. 297 00:27:05,840 --> 00:27:08,460 And that has to be 0. 298 00:27:08,460 --> 00:27:10,890 And this involves only the first product of 299 00:27:10,890 --> 00:27:12,390 the direction cosine. 300 00:27:12,390 --> 00:27:17,180 So to make it come out 0 when we add up the magnitudes, we 301 00:27:17,180 --> 00:27:20,800 will get the sign of the direction cosine. 302 00:27:20,800 --> 00:27:23,750 So this is a much more powerful relation. 303 00:27:23,750 --> 00:27:28,480 And similarly, the product of the coefficients in the first 304 00:27:28,480 --> 00:27:31,470 and the third row have to add up to 0. 305 00:27:31,470 --> 00:27:34,620 And the second and third row have to be 0. 306 00:27:34,620 --> 00:27:38,550 So there are three different relations we can write between 307 00:27:38,550 --> 00:27:42,970 products of corresponding coefficients in the rows. 308 00:27:45,890 --> 00:27:50,900 So to sum up in words, the sum of the 309 00:27:50,900 --> 00:28:00,795 corresponding elements-- 310 00:28:05,990 --> 00:28:07,350 of the product of-- 311 00:28:13,410 --> 00:28:32,190 in any pair of rows of Cij must be 0. 312 00:28:38,770 --> 00:28:41,650 But we're not done yet. 313 00:28:41,650 --> 00:28:49,680 If we look at the columns in this array, this represents 314 00:28:49,680 --> 00:28:57,590 the components of x1 relative to x1 prime, 315 00:28:57,590 --> 00:28:59,410 x2 prime, x3 prime. 316 00:28:59,410 --> 00:29:04,410 And these terms here represent the components of x2 relative 317 00:29:04,410 --> 00:29:08,110 to x1 prime, x2 prime, and x3 prime. 318 00:29:08,110 --> 00:29:11,410 And for similar reasons, the dot product of those two 319 00:29:11,410 --> 00:29:13,090 vectors has to be 0. 320 00:29:13,090 --> 00:29:20,440 So we can say that in addition, the sum of any sum 321 00:29:20,440 --> 00:29:36,480 of pairs of corresponding coefficients in any pair of 322 00:29:36,480 --> 00:29:44,980 columns must be 0. 323 00:29:53,730 --> 00:29:57,330 So we're working here on the direct matrix of the 324 00:29:57,330 --> 00:29:58,230 transformation. 325 00:29:58,230 --> 00:30:02,970 We've seen that the reverse relation, the inverse matrix 326 00:30:02,970 --> 00:30:07,830 of Cij is Cij transpose. 327 00:30:07,830 --> 00:30:13,290 And therefore the inverse matrix has to have this same 328 00:30:13,290 --> 00:30:16,840 relationship that the products of terms and rows or columns, 329 00:30:16,840 --> 00:30:18,990 any pair of rows or columns has to be 0. 330 00:30:25,110 --> 00:30:30,660 Now, there's one other pair of relations among the 331 00:30:30,660 --> 00:30:34,260 coefficients, which is not quite so 332 00:30:34,260 --> 00:30:36,050 geometrically obvious. 333 00:30:36,050 --> 00:30:37,920 And I won't attempt to prove it. 334 00:30:37,920 --> 00:30:40,240 I'll just state it. 335 00:30:40,240 --> 00:30:43,820 I said a moment ago that these are unitary matrices. 336 00:30:43,820 --> 00:30:48,160 The determinant of the coefficient Cij 337 00:30:48,160 --> 00:30:51,270 then has to be unity. 338 00:30:51,270 --> 00:30:59,190 But interestingly, it will be plus 1 if one goes from a 339 00:30:59,190 --> 00:31:01,925 right-handed system to a right-handed system. 340 00:31:10,890 --> 00:31:14,770 That is to say the set of axes x1, x2, x3 might be 341 00:31:14,770 --> 00:31:15,730 right-handed. 342 00:31:15,730 --> 00:31:19,890 And if the new coordinate system x1 prime, x2 prime, x3 343 00:31:19,890 --> 00:31:22,450 prime is also right-handed, then the determinant of 344 00:31:22,450 --> 00:31:24,850 coefficients is plus 1. 345 00:31:24,850 --> 00:31:27,530 On the other hand, if one goes from a right-handed system to 346 00:31:27,530 --> 00:31:30,960 a left-handed reference system or from a left-handed one to a 347 00:31:30,960 --> 00:31:33,910 right-handed coefficient, then, interestingly the 348 00:31:33,910 --> 00:31:38,040 determinant of coefficients is minus 1. 349 00:31:38,040 --> 00:31:40,380 So the determinant of the matrix of the transformation 350 00:31:40,380 --> 00:31:42,700 is plus 1 if you go to coordinate 351 00:31:42,700 --> 00:31:44,900 system of the same chirality. 352 00:31:44,900 --> 00:31:48,190 It's equal to minus 1 if you go to a coordinate system of 353 00:31:48,190 --> 00:31:49,440 changed chirality. 354 00:31:55,080 --> 00:31:59,580 All right, so to repeat something I said at the outset 355 00:31:59,580 --> 00:32:06,710 but which you now probably truly believe, the elements in 356 00:32:06,710 --> 00:32:10,140 the direction cosine scheme that get you from one 357 00:32:10,140 --> 00:32:13,460 coordinate system to another have lots of inter-relations. 358 00:32:13,460 --> 00:32:17,140 And all of these coefficients are not independent. 359 00:32:17,140 --> 00:32:19,036 There are these relations that couple them. 360 00:32:23,230 --> 00:32:24,480 How are we doing on time? 361 00:32:39,120 --> 00:32:47,080 We mentioned last time that a large collection of physical 362 00:32:47,080 --> 00:32:52,060 properties of materials are properties that relate a pair 363 00:32:52,060 --> 00:32:54,730 of vectors. 364 00:32:54,730 --> 00:33:00,520 So let me, to make this specific, talk about a 365 00:33:00,520 --> 00:33:03,205 particular physical property, electrical conductivity. 366 00:33:11,110 --> 00:33:16,130 And electrical conductivity relates a current density 367 00:33:16,130 --> 00:33:37,715 vector, and that it charge per unit area per unit time to an 368 00:33:37,715 --> 00:33:40,930 applied vector, and that vector is the 369 00:33:40,930 --> 00:33:42,180 electric field vector. 370 00:33:49,980 --> 00:33:55,250 And the electric field has units of volts per unit 371 00:33:55,250 --> 00:33:57,870 length, so volts per meter in MKS. 372 00:34:03,530 --> 00:34:09,850 And provided the electric field that's supplied is not 373 00:34:09,850 --> 00:34:15,440 too strong, it turns out that every component of the current 374 00:34:15,440 --> 00:34:21,280 flow is given by a linear combination of every component 375 00:34:21,280 --> 00:34:24,139 of the applied electric field. 376 00:34:24,139 --> 00:34:29,170 So the flow of current along x1 will be given by a 377 00:34:29,170 --> 00:34:35,030 proportionality constant, an element sigma 1 1 times the x1 378 00:34:35,030 --> 00:34:36,659 component of the electric field. 379 00:34:36,659 --> 00:34:39,159 Let me write it out the first couple of times we do this. 380 00:34:39,159 --> 00:34:44,600 Sigma 1 1 times E1 plus sigma 1 2 times E2 plus 381 00:34:44,600 --> 00:34:48,739 sigma 1 3 times E3. 382 00:34:48,739 --> 00:34:55,340 J2 will be sigma 2 1 times E1 plus sigma 2 2 times E2 plus 383 00:34:55,340 --> 00:34:59,560 sigma 2 3 times E3. 384 00:34:59,560 --> 00:35:06,540 And J3 will be equal to sigma 3 1 times E1 plus sigma 3 2 385 00:35:06,540 --> 00:35:10,960 times E2 plus sigma 3 3 times E3. 386 00:35:14,090 --> 00:35:19,370 Looks formally like the relation between unit vectors 387 00:35:19,370 --> 00:35:21,660 that define a coordinate system. 388 00:35:21,660 --> 00:35:24,340 Number of subscripts is the same, but actually this is 389 00:35:24,340 --> 00:35:25,700 something that's completely different. 390 00:35:25,700 --> 00:35:28,900 It's dealing with vectors that have some physical 391 00:35:28,900 --> 00:35:30,150 significance. 392 00:35:35,060 --> 00:35:39,620 So in compact reduced subscript notation, this is 393 00:35:39,620 --> 00:35:41,646 the definition of electrical conductivity. 394 00:35:48,440 --> 00:35:54,280 This matrix that relates the electric field vector to the 395 00:35:54,280 --> 00:35:58,440 current density vector is said to be a tensor 396 00:35:58,440 --> 00:35:59,760 of the second rank. 397 00:36:09,560 --> 00:36:11,800 OK, tensor. 398 00:36:11,800 --> 00:36:13,550 First thing you might say, why do you call 399 00:36:13,550 --> 00:36:14,350 it a tensor, dummy? 400 00:36:14,350 --> 00:36:15,030 It's a matrix. 401 00:36:15,030 --> 00:36:17,510 It's a plain old matrix. 402 00:36:17,510 --> 00:36:21,060 There's a subtle but very important difference. 403 00:36:21,060 --> 00:36:24,270 A tensor is a matrix with an attitude. 404 00:36:24,270 --> 00:36:29,040 And I'll make the distinction clear a little bit later on. 405 00:36:29,040 --> 00:36:33,350 But there are tensors also of higher rank. 406 00:36:33,350 --> 00:36:37,810 These expressions where summation over repeated 407 00:36:37,810 --> 00:36:43,830 subscripts is implied can hide, as I indicated last 408 00:36:43,830 --> 00:36:47,080 time, some absolutely horrendous polynomials. 409 00:36:47,080 --> 00:36:55,020 But tensor at very least is a term that makes the faces of 410 00:36:55,020 --> 00:36:59,200 all who hear it pale, and makes the knees of even the 411 00:36:59,200 --> 00:37:03,930 very strong to weaken. 412 00:37:03,930 --> 00:37:07,010 And in case you don't believe that, I'll show you what I 413 00:37:07,010 --> 00:37:09,155 have to wear whenever I give these lectures. 414 00:37:09,155 --> 00:37:11,890 And consequently it's kind of scuzzy and worn out. 415 00:37:11,890 --> 00:37:13,790 But I have to put on these knee braces 416 00:37:13,790 --> 00:37:15,100 from wobbling braces. 417 00:37:15,100 --> 00:37:16,620 And you can see what it says on here. 418 00:37:16,620 --> 00:37:23,300 "Tensor." So that's a consequence of this 419 00:37:23,300 --> 00:37:25,940 frightening definition that we've just made. 420 00:37:28,850 --> 00:37:31,890 Let me next set the stage for what we ought to do next. 421 00:37:37,430 --> 00:37:41,220 E sub j represents the components of an electric 422 00:37:41,220 --> 00:37:44,980 field, x1, x2, x3, in a first coordinate system. 423 00:37:50,640 --> 00:37:55,570 ji represent the components of the current flow in a 424 00:37:55,570 --> 00:37:57,480 coordinate system, x1, x2, x3. 425 00:38:00,660 --> 00:38:04,870 If we were to change coordinate system for any 426 00:38:04,870 --> 00:38:08,530 reason, these three numbers would wink on and off. 427 00:38:08,530 --> 00:38:09,970 Some might go negative. 428 00:38:09,970 --> 00:38:12,040 The magnitudes would change. 429 00:38:12,040 --> 00:38:15,920 And as a result, the components of the current flow 430 00:38:15,920 --> 00:38:17,790 would have to do the same thing. 431 00:38:17,790 --> 00:38:19,950 Because the components of these vectors, without 432 00:38:19,950 --> 00:38:24,860 changing anything physically, have to change their numerical 433 00:38:24,860 --> 00:38:31,170 values if we refer them to a new set of reference axes. 434 00:38:31,170 --> 00:38:35,480 If we change coordinate system and these numbers change, and 435 00:38:35,480 --> 00:38:38,820 if we change coordinate system, these numbers change, 436 00:38:38,820 --> 00:38:41,290 we're still applying field in the same direction. 437 00:38:41,290 --> 00:38:43,540 The current still flows in the same direction. 438 00:38:43,540 --> 00:38:46,770 But the components we use to define 439 00:38:46,770 --> 00:38:48,940 these two vectors change. 440 00:38:48,940 --> 00:38:52,550 And it follows just algebraically, the elements of 441 00:38:52,550 --> 00:38:57,940 the tensor have to change and link into different values. 442 00:38:57,940 --> 00:39:00,590 It follows automatically. 443 00:39:00,590 --> 00:39:04,040 So a question, then, is that if we have a coordinate 444 00:39:04,040 --> 00:39:07,990 system, x1, x2, x3, and we change it into a new 445 00:39:07,990 --> 00:39:15,056 coordinate system, x1 prime, x2 prime, x3 prime, then j sub 446 00:39:15,056 --> 00:39:19,690 i changes to some new values, j sub i prime. 447 00:39:19,690 --> 00:39:24,960 E sub j changes to some new values, E sub j prime. 448 00:39:24,960 --> 00:39:29,360 And therefore, of necessity, sigma i j, the conductivity 449 00:39:29,360 --> 00:39:35,690 tensor, has to change to new values sigma ij prime. 450 00:39:35,690 --> 00:39:38,730 So I'll let you rest up to brace yourself for this. 451 00:39:38,730 --> 00:39:44,190 The question is, how can we get sigma ij prime, the nine 452 00:39:44,190 --> 00:39:47,360 elements of the tensor in the new coordinate system, in 453 00:39:47,360 --> 00:39:50,660 terms of the direction cosine scheme that defines this 454 00:39:50,660 --> 00:39:55,160 transformation and in terms of the elements of the original 455 00:39:55,160 --> 00:39:57,710 conductivity tensor? 456 00:39:57,710 --> 00:40:01,300 And this, my friends, is what makes a tensor a tensor and 457 00:40:01,300 --> 00:40:03,440 not a matrix. 458 00:40:03,440 --> 00:40:09,850 I can write a matrix for you, a really lovely matrix. 459 00:40:09,850 --> 00:40:13,560 Let's put in some elements here. 460 00:40:13,560 --> 00:40:21,750 Let's put in 6.2, square root of minus 1e, and 23. 461 00:40:21,750 --> 00:40:30,590 And as other elements, I'll put in pi 23.4, 6, and 0. 462 00:40:30,590 --> 00:40:32,830 It's a perfectly good matrix. 463 00:40:32,830 --> 00:40:36,500 It's just an array of numbers, any numbers, real or 464 00:40:36,500 --> 00:40:38,540 imaginary, or whatever I like. 465 00:40:38,540 --> 00:40:39,790 So this is a matrix. 466 00:40:42,090 --> 00:40:51,430 What a tensor is, is a matrix for which a law of 467 00:40:51,430 --> 00:40:53,055 transformation is defined. 468 00:41:04,700 --> 00:41:06,880 And that's what makes a tensor a tensor. 469 00:41:12,860 --> 00:41:16,420 What does it mean to take this two-by-four matrix that I just 470 00:41:16,420 --> 00:41:16,890 wrote down? 471 00:41:16,890 --> 00:41:20,240 How do I transform that to a different coordinate system? 472 00:41:20,240 --> 00:41:22,660 It's meaningless, just an array of numbers. 473 00:41:22,660 --> 00:41:25,270 It's an array of numbers that has some useful properties, 474 00:41:25,270 --> 00:41:27,230 like matrix multiplication and the like. 475 00:41:27,230 --> 00:41:30,910 But to talk about transformation of this set of 476 00:41:30,910 --> 00:41:33,430 four ridiculous numbers to a new coordinate system is 477 00:41:33,430 --> 00:41:35,590 something that's absolutely meaningless. 478 00:41:35,590 --> 00:41:38,910 Not so for something like conductivity or the 479 00:41:38,910 --> 00:41:43,310 piezoelectric moduli or the elastic constants. 480 00:41:43,310 --> 00:41:44,550 These change their values. 481 00:41:44,550 --> 00:41:47,130 There's a law of transformation when we go from 482 00:41:47,130 --> 00:41:49,380 one Cartesian reference system to another. 483 00:41:52,960 --> 00:41:58,895 So what we will do when and if you return is to derive a law 484 00:41:58,895 --> 00:42:02,440 for transformation for second-rank tensors, and then, 485 00:42:02,440 --> 00:42:06,810 by implication, look at higher-rank tensors and decide 486 00:42:06,810 --> 00:42:09,920 how they would transform. 487 00:42:09,920 --> 00:42:11,920 But why would you want to do this? 488 00:42:11,920 --> 00:42:14,860 Why would you want to muck things up and have to worry 489 00:42:14,860 --> 00:42:16,530 about transforming these numbers? 490 00:42:16,530 --> 00:42:20,200 Well, let me give you just one simple example. 491 00:42:20,200 --> 00:42:25,790 Suppose we had conductivity of a plate, of a crystal. 492 00:42:25,790 --> 00:42:26,880 And what would you do? 493 00:42:26,880 --> 00:42:34,140 You'd measure it relative to a set of axes, which, if you 494 00:42:34,140 --> 00:42:36,020 have a little fragment of crystal, you have 495 00:42:36,020 --> 00:42:37,100 no reference system. 496 00:42:37,100 --> 00:42:44,590 So say that the axes x of i are taken relative to the 497 00:42:44,590 --> 00:42:51,700 lattice constants of the material, so relative to the 498 00:42:51,700 --> 00:42:58,540 edges of the unit cell, possibly. 499 00:42:58,540 --> 00:43:01,710 Then you decide that this material really has some 500 00:43:01,710 --> 00:43:05,890 useful properties, and you would like to cut a piece out 501 00:43:05,890 --> 00:43:13,270 of it so that you get a plate for which the maximum 502 00:43:13,270 --> 00:43:19,650 conductivity in that plate is in a direction 503 00:43:19,650 --> 00:43:22,330 normal to the plate. 504 00:43:22,330 --> 00:43:24,450 So you know just what sort of plate you want to cut out. 505 00:43:24,450 --> 00:43:26,480 You know what the direction cosines are. 506 00:43:26,480 --> 00:43:30,820 But once you've cut a plate from the crystal, the tensor 507 00:43:30,820 --> 00:43:34,860 relative to the old axes, x1, x2, x3, is not going to be 508 00:43:34,860 --> 00:43:35,970 terribly useful. 509 00:43:35,970 --> 00:43:39,720 You're going to want to find the tensor relative to this as 510 00:43:39,720 --> 00:43:44,040 one set of axes, and these perhaps as a new set of axes 511 00:43:44,040 --> 00:43:47,570 within the plane of the plate. 512 00:43:47,570 --> 00:43:48,730 So there's a good example. 513 00:43:48,730 --> 00:43:54,350 Cut a piece from a crystal and cut that piece so that the 514 00:43:54,350 --> 00:43:58,180 extreme values are along x, y, and z for the 515 00:43:58,180 --> 00:43:59,420 new coordinate system. 516 00:43:59,420 --> 00:44:02,840 Then you will be faced with the necessity of transforming 517 00:44:02,840 --> 00:44:07,330 the tensor from one coordinate system to another one. 518 00:44:07,330 --> 00:44:10,150 Or you might measure the thermal conductivity tensor. 519 00:44:10,150 --> 00:44:13,120 You might want to cut a rod out of the material so that 520 00:44:13,120 --> 00:44:16,930 the maximum conductivity or the minimum thermal 521 00:44:16,930 --> 00:44:19,380 conductivity is along the direction of the rod. 522 00:44:19,380 --> 00:44:22,310 You might want to use that as a push rod to hold a sample in 523 00:44:22,310 --> 00:44:26,360 position and not have it be a big heat sink for the 524 00:44:26,360 --> 00:44:30,340 temperature that's inside of your sample chamber. 525 00:44:30,340 --> 00:44:33,480 So I've hopefully convinced you that there are lots of 526 00:44:33,480 --> 00:44:38,010 cases where it would be necessary and convenient to 527 00:44:38,010 --> 00:44:42,100 transform the tensor that describes a property to a new 528 00:44:42,100 --> 00:44:43,350 coordinate system. 529 00:44:48,270 --> 00:44:50,770 All right, so let us take our break now. 530 00:44:50,770 --> 00:44:55,130 Some internal clock always tells me when it's five of the 531 00:44:55,130 --> 00:44:57,560 hour, unless I get really excited about something. 532 00:44:57,560 --> 00:44:59,360 And it is indeed that time now. 533 00:44:59,360 --> 00:45:00,610 So let's stop.