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.
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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
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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.
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00:03:13,860 --> 00:03:20,890
And if we put in magnitude of P
equal to 1, it follows that
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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
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00:04:35,980 --> 00:04:39,690
coordinate system that we're
using as our reference system.
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00:04:39,690 --> 00:04:42,970
So the next question we might
ask is, suppose we change the
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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.
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00:05:03,890 --> 00:05:06,530
And this is my vector P.
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00:05:06,530 --> 00:05:12,950
And I change x1 to some new
value, x1 prime, and change x2
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00:05:12,950 --> 00:05:16,170
to some new orientation,
x2 prime.
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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.
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00:05:32,050 --> 00:05:36,440
If I change the direction of x2
and draw a perpendicular to
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00:05:36,440 --> 00:05:39,100
x2, this would be P2 prime.
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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,
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00:05:45,380 --> 00:05:49,850
P2, P3, are going to change to
some new values, P1 prime, P2
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00:05:49,850 --> 00:05:51,100
prime, P3 prime.
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00:05:57,060 --> 00:05:57,280
OK.
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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.
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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
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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.
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00:06:57,530 --> 00:07:01,080
So I'm saying that this is going
to be the sum of the
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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
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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.
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00:08:34,520 --> 00:08:35,978
I don't want a prime on that.
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00:08:38,900 --> 00:08:45,110
So I'm going to say now that the
component of P along the
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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.