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OK, so anyway,
let's get started.
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00:00:27 --> 00:00:31
So,
the first unit of the class,
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so basically I'm going to go
over the first half of the class
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today,
and the second half of the
11
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class on Tuesday just because we
have to start somewhere.
12
00:00:41 --> 00:00:48
So, the first things that we
learned about in this class were
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00:00:48 --> 00:00:54
vectors, and how to do
dot-product of vectors.
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00:00:54 --> 00:01:01
So, remember the formula that A
dot B is the sum of ai times bi.
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00:01:01 --> 00:01:05
And, geometrically,
it's length A times length B
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00:01:05 --> 00:01:08
times the cosine of the angle
between them.
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00:01:08 --> 00:01:11
And, in particular,
we can use this to detect when
18
00:01:11 --> 00:01:14
two vectors are perpendicular.
That's when their dot product
19
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is zero.
And, we can use that to measure
20
00:01:17 --> 00:01:21
angles between vectors by
solving for cosine in this.
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Hopefully, at this point,
this looks a lot easier than it
22
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used to a few months ago.
So, hopefully at this point,
23
00:01:28 --> 00:01:32
everyone has this kind of
formula memorized and has some
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00:01:32 --> 00:01:35
reasonable understanding of
that.
25
00:01:35 --> 00:01:41
But, if you have any questions,
now is the time.
26
00:01:41 --> 00:01:45
No?
Good.
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00:01:45 --> 00:01:55
Next we learned how to also do
cross product of vectors in
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00:01:55 --> 00:02:06
space -- -- and remember,
we saw how to use that to find
29
00:02:06 --> 00:02:10
area of, say,
a triangle or a parallelogram
30
00:02:10 --> 00:02:14
in space because the length of
the cross product is equal to
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00:02:14 --> 00:02:17
the area of a parallelogram
formed by the vectors a and b.
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00:02:17 --> 00:02:25
And, we can also use that to
find a vector perpendicular to
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00:02:25 --> 00:02:28
two given vectors,
A and B.
34
00:02:28 --> 00:02:33
And so, in particular,
that comes in handy when we are
35
00:02:33 --> 00:02:42
looking for the equation of a
plane because we've seen -- So,
36
00:02:42 --> 00:02:49
the next topic would be
equations of planes.
37
00:02:49 --> 00:02:55
And, we've seen that when you
put the equation of a plane in
38
00:02:55 --> 00:02:59
the form ax by cz = d,
well, 00:03:03
b, c> in there is actually
the normal vector to the plane,
40
00:03:03 --> 00:03:07
or some normal vector to the
plane.
41
00:03:07 --> 00:03:11
So, typically,
we use cross product to find
42
00:03:11 --> 00:03:16
plane equations.
OK, is that still reasonably
43
00:03:16 --> 00:03:21
familiar to everyone?
Yes, very good.
44
00:03:21 --> 00:03:26
OK, we've also seen how to look
at equations of lines,
45
00:03:26 --> 00:03:31
and those were of a slightly
different nature because we've
46
00:03:31 --> 00:03:35
been doing them as parametric
equations.
47
00:03:35 --> 00:03:42
So, typically we had equations
of a form, maybe x equals some
48
00:03:42 --> 00:03:47
constant times t,
y equals constant plus constant
49
00:03:47 --> 00:03:53
times t.
z equals constant plus constant
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00:03:53 --> 00:04:02
times t where these terms here
correspond to some point on the
51
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line.
And, these coefficients here
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00:04:06 --> 00:04:11
correspond to a vector parallel
to the line.
53
00:04:11 --> 00:04:19
That's the velocity of the
moving point on the line.
54
00:04:19 --> 00:04:23
And, well,
we've learned in particular how
55
00:04:23 --> 00:04:29
to find where a line intersects
a plane by plugging in the
56
00:04:29 --> 00:04:34
parametric equation into the
equation of a plane.
57
00:04:34 --> 00:04:43
We've learned more general
things about parametric
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00:04:43 --> 00:04:48
equations of curves.
So, there are these infamous
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00:04:48 --> 00:04:51
problems in particular where you
have these rotating wheels and
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points on them,
and you have to figure out,
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00:04:53 --> 00:04:57
what's the position of a point?
And, the general principle of
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00:04:57 --> 00:05:01
those is that you want to
decompose the position vector
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into a sum of simpler things.
OK, so if you have a point on a
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00:05:05 --> 00:05:08
wheel that's itself moving and
something else,
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00:05:08 --> 00:05:11
then you might want to first
figure out the position of a
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center of a wheel than find the
angle by which the wheel has
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turned,
and then get to the position of
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a moving point by adding
together simpler vectors.
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So, the general principle is
really to try to find one
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parameter that will let us
understand what has happened,
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00:05:30 --> 00:05:36
and then decompose the motion
into a sum of simpler effect.
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00:05:36 --> 00:05:54
So, we want to decompose the
position vector into a sum of
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00:05:54 --> 00:06:02
simpler vectors.
OK, so maybe now we are getting
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a bit out of some people's
comfort zone,
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00:06:05 --> 00:06:12
but hopefully it's not too bad.
Do you have any general
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00:06:12 --> 00:06:20
questions about how one would go
about that, or,
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00:06:20 --> 00:06:24
yes?
Sorry? What about it?
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00:06:24 --> 00:06:25
Parametric descriptions of a
plane,
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00:06:25 --> 00:06:28
so we haven't really done that
because you would need two
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00:06:28 --> 00:06:31
parameters to parameterize a
plane just because it's a two
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00:06:31 --> 00:06:35
dimensional object.
So, we have mostly focused on
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00:06:35 --> 00:06:40
the use of parametric equations
just for one dimensional
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00:06:40 --> 00:06:42
objects, lines,
and curves.
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So,
you won't need to know about
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00:06:45 --> 00:06:47
parametric descriptions of
planes on a final,
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00:06:47 --> 00:06:51
but if you really wanted to,
you would think of defining a
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00:06:51 --> 00:06:55
point on a plane as starting
from some given point.
88
00:06:55 --> 00:06:57
Then you have two vectors given
on the plane.
89
00:06:57 --> 00:07:00
And then, you would add a
multiple of each of these
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00:07:00 --> 00:07:04
vectors to your starting point.
But see, the difficulty is to
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00:07:04 --> 00:07:08
convert from that to the usual
equation of a plane,
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00:07:08 --> 00:07:11
you would still have to go back
to this cross product method,
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00:07:11 --> 00:07:15
and so on.
So, it is possible to represent
94
00:07:15 --> 00:07:19
a plane, or, in general,
a surface in parametric form.
95
00:07:19 --> 00:07:23
But, very often,
that's not so useful.
96
00:07:23 --> 00:07:28
Yes?
How do you parametrize an
97
00:07:28 --> 00:07:31
ellipse in space?
Well, that depends on how it's
98
00:07:31 --> 00:07:34
given to you.
But, OK, let's just do an
99
00:07:34 --> 00:07:38
example.
Say that I give you an ellipse
100
00:07:38 --> 00:07:42
in space as maybe the more,
well, one exciting way to
101
00:07:42 --> 00:07:45
parameterize an ellipse in space
is maybe the intersection of a
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00:07:45 --> 00:07:49
cylinder with a slanted plane.
That's the kind of situations
103
00:07:49 --> 00:07:52
where you might end up with an
ellipse.
104
00:07:52 --> 00:07:58
OK, so if I tell you that maybe
I'm intersecting a cylinder with
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00:07:58 --> 00:08:03
equation x squared plus y
squared equals a squared with a
106
00:08:03 --> 00:08:09
slanted plane to get,
I messed up my picture,
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00:08:09 --> 00:08:13
to get this ellipse of
intersection,
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00:08:13 --> 00:08:14
so, of course you'd need the
equation of a plane.
109
00:08:14 --> 00:08:18
And, let's say that this plane
is maybe given to you.
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00:08:18 --> 00:08:23
Or, you can switch it to form
where you can get z as a
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00:08:23 --> 00:08:29
function of x and y.
So, maybe it would be z equals,
112
00:08:29 --> 00:08:33
I've already used a;
I need to use a new letter.
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00:08:33 --> 00:08:41
Let's say c1x c2y plus d,
whatever, something like that.
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00:08:41 --> 00:08:45
So, what I would do is first I
would look at what my ellipse
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00:08:45 --> 00:08:49
does in the directions in which
I understand it the best.
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00:08:49 --> 00:08:53
And, those directions would be
probably the xy plane.
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00:08:53 --> 00:08:56
So, I would look at the xy
coordinates.
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00:08:56 --> 00:09:02
Well, if I look at it from
above xy, my ellipse looks like
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00:09:02 --> 00:09:06
just a circle of radius a.
So, if I'm only concerned with
120
00:09:06 --> 00:09:10
x and y, presumably I can just
do it the usual way for a
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00:09:10 --> 00:09:13
circle.
x equals a cosine t.
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00:09:13 --> 00:09:20
y equals a sine t, OK?
And then, z would end up being
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00:09:20 --> 00:09:24
just, well, whatever the value
of z is to be on the slanted
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00:09:24 --> 00:09:29
plane above a given xy position.
So, in fact,
125
00:09:29 --> 00:09:38
it would end up being ac1
cosine t plus ac2 sine t plus d,
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00:09:38 --> 00:09:42
I guess.
OK, that's not a particularly
127
00:09:42 --> 00:09:44
elegant parameterization,
but that's the kind of thing
128
00:09:44 --> 00:09:47
you might end up with.
Now, in general,
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00:09:47 --> 00:09:50
when you have a curve in space,
it would rarely be the case
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00:09:50 --> 00:09:53
that you have to get a
parameterization from scratch
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00:09:53 --> 00:09:56
unless you are already being
told information about how it
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00:09:56 --> 00:09:58
looks in one of the coordinate
planes,
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00:09:58 --> 00:10:03
this kind of method.
Or, at least you'd have a lot
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00:10:03 --> 00:10:07
of information that would
quickly reduce to a plane
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00:10:07 --> 00:10:11
problem somehow.
Of course, I could also just
136
00:10:11 --> 00:10:16
give you some formulas and let
you figure out what's going on.
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00:10:16 --> 00:10:21
But, in general,
we've done more stuff with
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00:10:21 --> 00:10:25
plane curves.
With plane curves,
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00:10:25 --> 00:10:29
certainly there's interesting
things with all sorts of
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00:10:29 --> 00:10:32
mechanical gadgets that we can
study.
141
00:10:32 --> 00:10:39
OK, any other questions on that?
No?
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00:10:39 --> 00:10:45
OK, so let me move on a bit and
point out that with parametric
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00:10:45 --> 00:10:51
equations, we've looked also at
things like velocity and
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00:10:51 --> 00:10:55
acceleration.
So, the velocity vector is the
145
00:10:55 --> 00:10:59
derivative of a position vector
with respect to time.
146
00:10:59 --> 00:11:04
And, it's not to be confused
with speed, which is the
147
00:11:04 --> 00:11:08
magnitude of v.
So, the velocity vector is
148
00:11:08 --> 00:11:12
going to be always tangent to
the curve.
149
00:11:12 --> 00:11:14
And, its length will be the
speed.
150
00:11:14 --> 00:11:15
That's the geometric
interpretation.
151
00:11:15 --> 00:11:32
152
00:11:32 --> 00:11:37
So, just to provoke you,
I'm going to write,
153
00:11:37 --> 00:11:43
again, that formula that was
that v equals T hat ds dt.
154
00:11:43 --> 00:11:46
What do I mean by that?
If I have a curve,
155
00:11:46 --> 00:11:51
and I'm moving on the curve,
well, I have the unit tangent
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00:11:51 --> 00:11:56
vector which I think at the time
I used to draw in blue.
157
00:11:56 --> 00:11:59
But, blue has been abolished
since then.
158
00:11:59 --> 00:12:04
So, I'm going to draw it in red.
OK, so that's a unit vector
159
00:12:04 --> 00:12:09
that goes along the curve,
and then the actual velocity is
160
00:12:09 --> 00:12:11
going to be proportional to
that.
161
00:12:11 --> 00:12:15
And, what's the length?
Well, it's the speed.
162
00:12:15 --> 00:12:19
And, the speed is how much arc
length on the curve I go per
163
00:12:19 --> 00:12:22
unit time, which is why I'm
writing ds dt.
164
00:12:22 --> 00:12:30
That's another guy.
That's another of these guys
165
00:12:30 --> 00:12:34
for the speed,
OK?
166
00:12:34 --> 00:12:41
And, we've also learned about
acceleration,
167
00:12:41 --> 00:12:47
which is the derivative of
velocity.
168
00:12:47 --> 00:12:50
So, it's the second derivative
of a position vector.
169
00:12:50 --> 00:12:54
And, as an example of the kinds
of manipulations we can do,
170
00:12:54 --> 00:12:56
in class we've seen Kepler's
second law,
171
00:12:56 --> 00:13:03
which explains how if the
acceleration is parallel to the
172
00:13:03 --> 00:13:08
position vector,
then r cross v is going to be
173
00:13:08 --> 00:13:10
constant,
which means that the motion
174
00:13:10 --> 00:13:13
will be in an plane,
and you will sweep area at a
175
00:13:13 --> 00:13:16
constant rate.
So now, that is not in itself a
176
00:13:16 --> 00:13:19
topic for the exam,
but the kinds of methods of
177
00:13:19 --> 00:13:22
differentiating vector
quantities,
178
00:13:22 --> 00:13:25
applying the product rule to
take the derivative of a dot or
179
00:13:25 --> 00:13:28
cross product and so on are
definitely fair game.
180
00:13:28 --> 00:13:30
I mean, we've seen those on the
first exam.
181
00:13:30 --> 00:13:35
They were there,
and most likely they will be on
182
00:13:35 --> 00:13:39
the final.
OK, so I mean that's the extent
183
00:13:39 --> 00:13:44
to which Kepler's law comes up,
only just knowing the general
184
00:13:44 --> 00:13:47
type of manipulations and
proving things with vector
185
00:13:47 --> 00:13:52
quantities,
but not again the actual
186
00:13:52 --> 00:13:58
Kepler's law itself.
I skipped something.
187
00:13:58 --> 00:14:08
I skipped matrices,
determinants,
188
00:14:08 --> 00:14:18
and linear systems.
OK, so we've seen how to
189
00:14:18 --> 00:14:24
multiply matrices,
and how to write linear systems
190
00:14:24 --> 00:14:28
in matrix form.
So, remember,
191
00:14:28 --> 00:14:35
if you have a 3x3 linear system
in the usual sense,
192
00:14:35 --> 00:14:42
so,
you can write this in a matrix
193
00:14:42 --> 00:14:52
form where you have a 3x3 matrix
and you have an unknown column
194
00:14:52 --> 00:14:57
vector.
And, their matrix product
195
00:14:57 --> 00:15:01
should be some given column
vector.
196
00:15:01 --> 00:15:04
OK, so if you don't remember
how to multiply matrices,
197
00:15:04 --> 00:15:07
please look at the notes on
that again.
198
00:15:07 --> 00:15:12
And, also you should remember
how to invert a matrix.
199
00:15:12 --> 00:15:22
So, how did we invert matrices?
Let me just remind you very
200
00:15:22 --> 00:15:30
quickly.
So, I should say 2x2 or 3x3
201
00:15:30 --> 00:15:33
matrices.
Well, you need to have a square
202
00:15:33 --> 00:15:35
matrix to be able to find an
inverse.
203
00:15:35 --> 00:15:37
The method doesn't work,
doesn't make sense.
204
00:15:37 --> 00:15:40
Otherwise, then the concept of
inverse doesn't work.
205
00:15:40 --> 00:15:43
And, if it's larger than 3x3,
then we haven't seen that.
206
00:15:43 --> 00:15:50
So, let's say that I have a 3x3
matrix.
207
00:15:50 --> 00:16:00
What I will do is I will start
by forming the matrix of minors.
208
00:16:00 --> 00:16:09
So, remember that minors,
so, each entry is a 2x2
209
00:16:09 --> 00:16:20
determinant in the case of a 3x3
matrix formed by deleting one
210
00:16:20 --> 00:16:26
row and one column.
OK, so for example,
211
00:16:26 --> 00:16:30
to get the first minor,
especially in the upper left
212
00:16:30 --> 00:16:34
corner, I would delete the first
row, the first column.
213
00:16:34 --> 00:16:36
And, I would be left with this
2x2 determinant.
214
00:16:36 --> 00:16:38
I take this times that minus
this times that.
215
00:16:38 --> 00:16:41
I get a number that gives my
first minor.
216
00:16:41 --> 00:16:49
And then, same with the others.
Then, I flip signs according to
217
00:16:49 --> 00:16:56
this checkerboard pattern,
and that gives me the matrix of
218
00:16:56 --> 00:17:00
cofactors.
OK, so all it means is I'm just
219
00:17:00 --> 00:17:06
changing the signs of these four
entries and leaving the others
220
00:17:06 --> 00:17:10
alone.
And then, I take the transpose
221
00:17:10 --> 00:17:13
of that.
So, that means I read it
222
00:17:13 --> 00:17:16
horizontally and write it down
vertically.
223
00:17:16 --> 00:17:19
I swept the rows and the
columns.
224
00:17:19 --> 00:17:23
And then, I divide by the
inverse.
225
00:17:23 --> 00:17:28
Well, I divide by the
determinant of the initial
226
00:17:28 --> 00:17:30
matrix.
OK, so, of course,
227
00:17:30 --> 00:17:32
this is kind of very
theoretical, and I write it like
228
00:17:32 --> 00:17:34
this.
Probably it makes more sense to
229
00:17:34 --> 00:17:37
do it on an example.
I will let you work out
230
00:17:37 --> 00:17:42
examples, or bug your recitation
instructors so that they do one
231
00:17:42 --> 00:17:44
on Monday if you want to see
that.
232
00:17:44 --> 00:17:47
It's a fairly straightforward
method.
233
00:17:47 --> 00:17:50
You just have to remember the
steps.
234
00:17:50 --> 00:17:52
But, of course,
there's one condition,
235
00:17:52 --> 00:17:57
which is that the determinant
of a matrix has to be nonzero.
236
00:17:57 --> 00:17:59
So, in fact,
we've seen that,
237
00:17:59 --> 00:18:03
oh, there is still one board
left.
238
00:18:03 --> 00:18:12
We've seen that a matrix is
invertible -- -- exactly when
239
00:18:12 --> 00:18:19
its determinant is not zero.
And, if that's the case,
240
00:18:19 --> 00:18:24
then we can solve the linear
system, AX equals B by just
241
00:18:24 --> 00:18:30
setting X equals A inverse B.
That's going to be the only
242
00:18:30 --> 00:18:38
solution to our linear system.
Otherwise, well,
243
00:18:38 --> 00:18:52
AX equals B has either no
solution, or infinitely many
244
00:18:52 --> 00:19:01
solutions.
Yes?
245
00:19:01 --> 00:19:04
The determinant of a matrix
real quick?
246
00:19:04 --> 00:19:08
Well, I can do it that quickly
unless I start waving my hands
247
00:19:08 --> 00:19:12
very quickly,
but remember we've seen that
248
00:19:12 --> 00:19:15
you have a matrix,
a 3x3 matrix.
249
00:19:15 --> 00:19:18
Its determinant will be
obtained by doing an expansion
250
00:19:18 --> 00:19:20
with respect to,
well, your favorite.
251
00:19:20 --> 00:19:22
But usually,
we are doing it with respect to
252
00:19:22 --> 00:19:26
the first row.
So, we take this entry and
253
00:19:26 --> 00:19:31
multiply it by that determinant.
Then, we take that entry,
254
00:19:31 --> 00:19:35
multiply it by that determinant
but put a minus sign.
255
00:19:35 --> 00:19:38
And then, we take that entry
and multiply it by this
256
00:19:38 --> 00:19:41
determinant here,
and we put a plus sign for
257
00:19:41 --> 00:19:44
that.
OK, so maybe I should write it
258
00:19:44 --> 00:19:46
down.
That's actually the same
259
00:19:46 --> 00:19:48
formula that we are using for
cross products.
260
00:19:48 --> 00:19:50
Right, when we do cross
products, we are doing an
261
00:19:50 --> 00:19:53
expansion with respect to the
first row.
262
00:19:53 --> 00:19:57
That's a special case.
OK, I mean, do you still want
263
00:19:57 --> 00:19:59
to see it in more details,
or is that OK?
264
00:19:59 --> 00:20:12
Yes?
That's correct.
265
00:20:12 --> 00:20:16
So, if you do an expansion with
respect to any row or column,
266
00:20:16 --> 00:20:19
then you would use the same
signs that are in this
267
00:20:19 --> 00:20:22
checkerboard pattern there.
So, if you did an expansion,
268
00:20:22 --> 00:20:25
actually, so indeed,
maybe I should say,
269
00:20:25 --> 00:20:28
the more general way to
determine it is you take your
270
00:20:28 --> 00:20:31
favorite row or column,
and you just multiply the
271
00:20:31 --> 00:20:34
corresponding entries by the
corresponding cofactors.
272
00:20:34 --> 00:20:37
So, the signs are plus or minus
depending on what's in that
273
00:20:37 --> 00:20:38
diagram there.
Now, in practice,
274
00:20:38 --> 00:20:41
in this class,
again, all we need is to do it
275
00:20:41 --> 00:20:46
with respect to the first row.
So, don't worry about it too
276
00:20:46 --> 00:20:48
much.
OK, so, again,
277
00:20:48 --> 00:20:51
the way that we've officially
seen it in this class is just if
278
00:20:51 --> 00:20:59
you have a1,
a2, a3, b1, b2, b3, c1, c2, c3,
279
00:20:59 --> 00:21:06
so if the determinant is a1
times b2 b3, c2 c3,
280
00:21:06 --> 00:21:16
minus a2 b1 b3 c1 c3 plus a3 b1
b2 c1 c2.
281
00:21:16 --> 00:21:20
And, this minus is here
basically because of the minus
282
00:21:20 --> 00:21:27
in the diagram up there.
But, that's all we need to know.
283
00:21:27 --> 00:21:32
Yes?
How do you tell the difference
284
00:21:32 --> 00:21:34
between infinitely many
solutions or no solutions?
285
00:21:34 --> 00:21:37
That's a very good question.
So, in full generality,
286
00:21:37 --> 00:21:40
the answer is we haven't quite
seen a systematic method.
287
00:21:40 --> 00:21:43
So, you just have to try
solving and see if you can find
288
00:21:43 --> 00:21:46
a solution or not.
So, let me actually explain
289
00:21:46 --> 00:21:51
that more carefully.
So, what happens to these two
290
00:21:51 --> 00:21:56
situations when a is invertible
or not?
291
00:21:56 --> 00:21:57
So, remember,
in the linear system,
292
00:21:57 --> 00:22:01
you can think of a linear
system as asking you to find the
293
00:22:01 --> 00:22:05
intersection between three
planes because each equation is
294
00:22:05 --> 00:22:12
the equation of a plane.
So, Ax = B for a 3x3 system
295
00:22:12 --> 00:22:24
means that x should be in the
intersection of three planes.
296
00:22:24 --> 00:22:28
And then, we have two cases.
So, the case where the system
297
00:22:28 --> 00:22:33
is invertible corresponds to the
general situation where your
298
00:22:33 --> 00:22:37
three planes somehow all just
intersect in one point.
299
00:22:37 --> 00:22:41
And then, the situation where
the determinant,
300
00:22:41 --> 00:22:45
that's when the determinant is
not zero, you get just one
301
00:22:45 --> 00:22:48
point.
However, sometimes it will
302
00:22:48 --> 00:22:54
happen that all the planes are
parallel to the same direction.
303
00:22:54 --> 00:23:04
So, determinant a equals zero
means the three planes are
304
00:23:04 --> 00:23:11
parallel to a same vector.
And, in fact,
305
00:23:11 --> 00:23:14
you can find that vector
explicitly because that vector
306
00:23:14 --> 00:23:17
has to be perpendicular to all
the normals.
307
00:23:17 --> 00:23:22
So, at some point we saw other
subtle things about how to find
308
00:23:22 --> 00:23:26
the direction of this line
that's parallel to all the
309
00:23:26 --> 00:23:30
planes.
So, now, this can happen either
310
00:23:30 --> 00:23:34
with all three planes containing
the same line.
311
00:23:34 --> 00:23:36
You know, they can all pass
through the same axis.
312
00:23:36 --> 00:23:39
Or it could be that they have
somehow shifted with respect to
313
00:23:39 --> 00:23:44
each other.
And so, it might look like this.
314
00:23:44 --> 00:23:46
Then, the last one is actually
in front of that.
315
00:23:46 --> 00:23:52
So, see, the lines of
intersections between two of the
316
00:23:52 --> 00:23:55
planes,
so, here they all pass through
317
00:23:55 --> 00:23:57
the same line,
and here, instead,
318
00:23:57 --> 00:24:00
they intersect in one line
here,
319
00:24:00 --> 00:24:03
one line here,
and one line there.
320
00:24:03 --> 00:24:06
And, there's no triple
intersection.
321
00:24:06 --> 00:24:08
So, in general,
we haven't really seen how to
322
00:24:08 --> 00:24:13
decide between these two cases.
There's one important situation
323
00:24:13 --> 00:24:20
where we have seen we must be in
the first case that when we have
324
00:24:20 --> 00:24:26
a homogeneous system,
so that means if the right hand
325
00:24:26 --> 00:24:31
side is zero,
then,
326
00:24:31 --> 00:24:41
well, x equals zero is always a
solution.
327
00:24:41 --> 00:24:43
It's called the trivial
solution.
328
00:24:43 --> 00:24:50
It's the obvious one,
if you want.
329
00:24:50 --> 00:24:53
So, you know that,
and why is that?
330
00:24:53 --> 00:24:57
Well, that's because all of
your planes have to pass through
331
00:24:57 --> 00:25:00
the origin.
So, you must be in this case if
332
00:25:00 --> 00:25:04
you have a noninvertible system
where the right hand side is
333
00:25:04 --> 00:25:05
zero.
So, in that case,
334
00:25:05 --> 00:25:08
if the right hand side is zero,
there's two cases.
335
00:25:08 --> 00:25:12
Either the matrix is invertible.
Then, the only solution is the
336
00:25:12 --> 00:25:14
trivial one.
Or, if a matrix is not
337
00:25:14 --> 00:25:19
invertible, then you have
infinitely many solutions.
338
00:25:19 --> 00:25:23
If B is not zero,
then we haven't really seen how
339
00:25:23 --> 00:25:27
to decide.
We've just seen how to decide
340
00:25:27 --> 00:25:30
between one solution or
zero,infinitely many,
341
00:25:30 --> 00:25:33
but not how to decide between
these last two cases.
342
00:25:33 --> 00:25:42
Yes?
I think in principle,
343
00:25:42 --> 00:25:44
you would be able to,
but that's, well,
344
00:25:44 --> 00:25:48
I mean, that's a slightly
counterintuitive way of doing
345
00:25:48 --> 00:25:50
it.
I think it would probably work.
346
00:25:50 --> 00:25:55
Well, I'll let you figure it
out.
347
00:25:55 --> 00:25:59
OK, let me move on to the
second unit, maybe,
348
00:25:59 --> 00:26:03
because we've seen a lot of
stuff, or was there a quick
349
00:26:03 --> 00:26:05
question before that?
OK.
350
00:26:05 --> 00:26:41
351
00:26:41 --> 00:26:44
OK, so what was the second part
of the class about?
352
00:26:44 --> 00:26:47
Well, hopefully you kind of
vaguely remember that it was
353
00:26:47 --> 00:26:50
about functions of several
variables and their partial
354
00:26:50 --> 00:26:55
derivatives.
OK, so the first thing that
355
00:26:55 --> 00:27:04
we've seen is how to actually
view a function of two variables
356
00:27:04 --> 00:27:12
in terms of its graph and its
contour plot.
357
00:27:12 --> 00:27:15
So,
just to remind you very
358
00:27:15 --> 00:27:17
quickly,
if I have a function of two
359
00:27:17 --> 00:27:21
variables, x and y,
then the graph will be just the
360
00:27:21 --> 00:27:25
surface given by the equation z
equals f of xy.
361
00:27:25 --> 00:27:28
So, for each x and y,
I plot a point at height given
362
00:27:28 --> 00:27:30
with the value of the a
function.
363
00:27:30 --> 00:27:34
And then, the contour plot will
be the topographical map for
364
00:27:34 --> 00:27:37
this graph.
It will tell us,
365
00:27:37 --> 00:27:41
what are the various levels in
there?
366
00:27:41 --> 00:27:46
So, what it amounts to is we
slice the graph by horizontal
367
00:27:46 --> 00:27:50
planes, and we get a bunch of
curves which are the points at
368
00:27:50 --> 00:27:56
given height on the plot.
And, so we get all of these
369
00:27:56 --> 00:28:04
curves, and then we look at them
from above, and that gives us
370
00:28:04 --> 00:28:09
this map with a bunch of curves
on it.
371
00:28:09 --> 00:28:13
And, each of them has a number
next to it which tells us the
372
00:28:13 --> 00:28:16
value of a function there.
And, from that map, we can,
373
00:28:16 --> 00:28:19
of course, tell things about
where we might be able to find
374
00:28:19 --> 00:28:22
minima or maxima of our
function,
375
00:28:22 --> 00:28:30
and how it varies with respect
to x or y or actually in any
376
00:28:30 --> 00:28:40
direction at a given point.
So, now, the next thing that
377
00:28:40 --> 00:28:49
we've learned about is partial
derivatives.
378
00:28:49 --> 00:28:52
So, for a function of two
variables, there would be two of
379
00:28:52 --> 00:28:54
them.
There's f sub x which is
380
00:28:54 --> 00:28:58
partial f partial x,
and f sub y which is partial f
381
00:28:58 --> 00:29:00
partial y.
And, in terms of a graph,
382
00:29:00 --> 00:29:04
they correspond to slicing by a
plane that's parallel to one of
383
00:29:04 --> 00:29:07
the coordinate planes,
so that we either keep x
384
00:29:07 --> 00:29:10
constant,
or keep y constant.
385
00:29:10 --> 00:29:14
And, we look at the slope of a
graph to see the rate of change
386
00:29:14 --> 00:29:17
of f with respect to one
variable only when we hold the
387
00:29:17 --> 00:29:21
other one constant.
And so, we've seen in
388
00:29:21 --> 00:29:25
particular how to use that in
various places,
389
00:29:25 --> 00:29:29
but, for example,
for linear approximation we've
390
00:29:29 --> 00:29:34
seen that the change in f is
approximately equal to f sub x
391
00:29:34 --> 00:29:40
times the change in x plus f sub
y times the change in y.
392
00:29:40 --> 00:29:45
So, you can think of f sub x
and f sub y as telling you how
393
00:29:45 --> 00:29:49
sensitive the value of f is to
changes in x and y.
394
00:29:49 --> 00:29:59
So, this linear approximation
also tells us about the tangent
395
00:29:59 --> 00:30:07
plane to the graph of f.
In fact, when we turn this into
396
00:30:07 --> 00:30:16
an equality, that would mean
that we replace f by the tangent
397
00:30:16 --> 00:30:19
plane.
We've also learned various ways
398
00:30:19 --> 00:30:21
of, before I go on,
I should say,
399
00:30:21 --> 00:30:24
of course, we've seen these
also for functions of three
400
00:30:24 --> 00:30:28
variables, right?
So, we haven't seen how to plot
401
00:30:28 --> 00:30:32
them, and we don't really worry
about that too much.
402
00:30:32 --> 00:30:37
But, if you have a function of
three variables,
403
00:30:37 --> 00:30:42
you can do the same kinds of
manipulations.
404
00:30:42 --> 00:30:49
So, we've learned about
differentials and chain rules,
405
00:30:49 --> 00:30:57
which are a way of repackaging
these partial derivatives.
406
00:30:57 --> 00:31:00
So, the differential is just,
by definition,
407
00:31:00 --> 00:31:05
this thing called df which is f
sub x times dx plus f sub y
408
00:31:05 --> 00:31:09
times dy.
And, what we can do with it is
409
00:31:09 --> 00:31:14
just either plug values for
changes in x and y,
410
00:31:14 --> 00:31:17
and get approximation formulas,
or we can look at this in a
411
00:31:17 --> 00:31:21
situation where x and y will
depend on something else,
412
00:31:21 --> 00:31:26
and we get a chain rule.
So, for example,
413
00:31:26 --> 00:31:32
if f is a function of t time,
for example, and so is y,
414
00:31:32 --> 00:31:36
then we can find the rate of
change of f with respect to t
415
00:31:36 --> 00:31:43
just by dividing this by dt.
So, we get df dt equals f sub x
416
00:31:43 --> 00:31:48
dx dt plus f sub y dy dt.
We can also get other chain
417
00:31:48 --> 00:31:51
rules,
say, if x and y depend on more
418
00:31:51 --> 00:31:54
than one variable,
if you have a change of
419
00:31:54 --> 00:31:55
variables,
for example,
420
00:31:55 --> 00:31:58
x and y are functions of two
other guys that you call u and
421
00:31:58 --> 00:32:01
v,
then you can express dx and dy
422
00:32:01 --> 00:32:05
in terms of du and dv,
and plugging into df you will
423
00:32:05 --> 00:32:08
get the manner in which f
depends on u and v.
424
00:32:08 --> 00:32:11
So, that will give you formulas
for partial f partial u,
425
00:32:11 --> 00:32:14
and partial f partial v.
They look just like these guys
426
00:32:14 --> 00:32:19
except there's a lot of curly
d's instead of straight ones,
427
00:32:19 --> 00:32:21
and u's and v's in the
denominators.
428
00:32:21 --> 00:32:26
OK, so that lets us understand
rates of change.
429
00:32:26 --> 00:32:31
We've also seen yet another way
to package partial derivatives
430
00:32:31 --> 00:32:33
into not a differential,
but instead,
431
00:32:33 --> 00:32:37
a vector.
That's the gradient vector,
432
00:32:37 --> 00:32:41
and I'm sure it was quite
mysterious when we first saw it,
433
00:32:41 --> 00:32:45
but hopefully by now,
well, it should be less
434
00:32:45 --> 00:32:46
mysterious.
435
00:32:46 --> 00:33:07
436
00:33:07 --> 00:33:14
OK, so we've learned about the
gradient vector which is del f
437
00:33:14 --> 00:33:21
is a vector whose components are
just the partial derivatives.
438
00:33:21 --> 00:33:26
So, if I have a function of
just two variables,
439
00:33:26 --> 00:33:29
then it's just this.
And,
440
00:33:29 --> 00:33:37
so one observation that we've
made is that if you look at a
441
00:33:37 --> 00:33:44
contour plot of your function,
so maybe your function is zero,
442
00:33:44 --> 00:33:47
one, and two,
then the gradient vector is
443
00:33:47 --> 00:33:49
always perpendicular to the
contour plot,
444
00:33:49 --> 00:33:54
and always points towards
higher ground.
445
00:33:54 --> 00:34:02
OK, so the reason for that was
that if you take any direction,
446
00:34:02 --> 00:34:04
you can measure the directional
derivative,
447
00:34:04 --> 00:34:12
which means the rate of change
of f in that direction.
448
00:34:12 --> 00:34:20
So, given a unit vector, u,
which represents some
449
00:34:20 --> 00:34:24
direction,
so for example let's say I
450
00:34:24 --> 00:34:29
decide that I want to go in this
direction,
451
00:34:29 --> 00:34:32
and I ask myself,
how quickly will f change if I
452
00:34:32 --> 00:34:36
start from here and I start
moving towards that direction?
453
00:34:36 --> 00:34:38
Well, the answer seems to be,
it will start to increase a
454
00:34:38 --> 00:34:41
bit, and maybe at some point
later on something else will
455
00:34:41 --> 00:34:45
happen.
But at first, it will increase.
456
00:34:45 --> 00:34:48
So,
the directional derivative is
457
00:34:48 --> 00:34:53
what we've called f by ds in the
direction of this unit vector,
458
00:34:53 --> 00:34:56
and basically the only thing we
know to be able to compute it,
459
00:34:56 --> 00:35:00
the only thing we need is that
it's the dot product between the
460
00:35:00 --> 00:35:02
gradient and this vector u hat.
In particular,
461
00:35:02 --> 00:35:05
the directional derivatives in
the direction of I hat or j hat
462
00:35:05 --> 00:35:07
are just the usual partial
derivatives.
463
00:35:07 --> 00:35:12
That's what you would expect.
OK, and so now you see in
464
00:35:12 --> 00:35:15
particular if you try to go in a
direction that's perpendicular
465
00:35:15 --> 00:35:18
to the gradient,
then the directional derivative
466
00:35:18 --> 00:35:21
will be zero because you are
moving on the level curve.
467
00:35:21 --> 00:35:27
So, the value doesn't change,
OK?
468
00:35:27 --> 00:35:45
Questions about that?
Yes?
469
00:35:45 --> 00:35:49
Yeah, so let's see,
so indeed to look at more
470
00:35:49 --> 00:35:52
recent things,
if you are taking the flux
471
00:35:52 --> 00:35:55
through something given by an
equation,
472
00:35:55 --> 00:35:59
so, if you have a surface given
by an equation,
473
00:35:59 --> 00:36:05
say, f equals one.
So, say that you have a surface
474
00:36:05 --> 00:36:08
here or a curve given by an
equation,
475
00:36:08 --> 00:36:14
f equals constant,
then the normal vector to the
476
00:36:14 --> 00:36:19
surface is given by taking the
gradient of f.
477
00:36:19 --> 00:36:22
And that is,
in general, not a unit normal
478
00:36:22 --> 00:36:24
vector.
Now, if you wanted the unit
479
00:36:24 --> 00:36:28
normal vector to compute flux,
then you would just scale this
480
00:36:28 --> 00:36:30
guy down to unit length,
OK?
481
00:36:30 --> 00:36:33
So, if you wanted a unit
normal, that would be the
482
00:36:33 --> 00:36:37
gradient divided by its length.
However, for flux,
483
00:36:37 --> 00:36:40
that's still of limited
usefulness because you would
484
00:36:40 --> 00:36:42
still need to know about ds.
But, remember,
485
00:36:42 --> 00:36:46
we've seen a formula for flux
in terms of a non-unit normal
486
00:36:46 --> 00:36:52
vector, and n over n dot kdxdy.
So, indeed, this is how you
487
00:36:52 --> 00:36:58
could actually handle
calculations of flux through
488
00:36:58 --> 00:37:09
pretty much anything.
Any other questions about that?
489
00:37:09 --> 00:37:19
OK, so let me continue with a
couple more things we need to,
490
00:37:19 --> 00:37:25
so, we've seen how to do
min/max problems,
491
00:37:25 --> 00:37:33
in particular,
by looking at critical points.
492
00:37:33 --> 00:37:35
So, critical points,
remember, are the points where
493
00:37:35 --> 00:37:37
all the partial derivatives are
zero.
494
00:37:37 --> 00:37:40
So, if you prefer,
that's where the gradient
495
00:37:40 --> 00:37:45
vector is zero.
And, we know how to decide
496
00:37:45 --> 00:37:52
using the second derivative test
whether a critical point is
497
00:37:52 --> 00:37:57
going to be a local min,
a local max,
498
00:37:57 --> 00:38:02
or a saddle point.
Actually, we can't always quite
499
00:38:02 --> 00:38:05
decide because,
remember, we look at the second
500
00:38:05 --> 00:38:08
partials, and we compute this
quantity ac minus b squared.
501
00:38:08 --> 00:38:10
And, if it happens to be zero,
then actually we can't
502
00:38:10 --> 00:38:13
conclude.
But, most of the time we can
503
00:38:13 --> 00:38:16
conclude.
However, that's not all we need
504
00:38:16 --> 00:38:20
to look for an absolute global
maximum or minimum.
505
00:38:20 --> 00:38:23
For that, we also need to check
the boundary points,
506
00:38:23 --> 00:38:27
or look at the behavior of a
function, at infinity.
507
00:38:27 --> 00:38:38
So, we also need to check the
values of f at the boundary of
508
00:38:38 --> 00:38:46
its domain of definition or at
infinity.
509
00:38:46 --> 00:38:48
Just to give you an example
from single variable calculus,
510
00:38:48 --> 00:38:51
if you are trying to find the
minimum and the maximum of f of
511
00:38:51 --> 00:38:55
x equals x squared,
well, you'll find quickly that
512
00:38:55 --> 00:38:57
the minimum is at zero where x
squared is zero.
513
00:38:57 --> 00:39:00
If you are looking for the
maximum, you better not just
514
00:39:00 --> 00:39:02
look at the derivative because
you won't find it that way.
515
00:39:02 --> 00:39:05
However, if you think for a
second, you'll see that if x
516
00:39:05 --> 00:39:08
becomes very large,
then the function increases to
517
00:39:08 --> 00:39:10
infinity.
And, similarly,
518
00:39:10 --> 00:39:14
if you try to find the minimum
and the maximum of x squared
519
00:39:14 --> 00:39:17
when x varies only between one
and two,
520
00:39:17 --> 00:39:19
well, you won't find the
critical point,
521
00:39:19 --> 00:39:21
but you'll still find that the
smallest value of x squared is
522
00:39:21 --> 00:39:24
when x is at one,
and the largest is at x equals
523
00:39:24 --> 00:39:26
two.
And, all this business about
524
00:39:26 --> 00:39:29
boundaries and infinity is
exactly the same stuff,
525
00:39:29 --> 00:39:31
but with more than one
variable.
526
00:39:31 --> 00:39:37
It's just the story that maybe
the minimum and the maximum are
527
00:39:37 --> 00:39:42
not quite visible,
but they are at the edges of a
528
00:39:42 --> 00:39:48
domain we are looking at.
Well, in the last three
529
00:39:48 --> 00:39:55
minutes, I will just write down
a couple more things we've seen
530
00:39:55 --> 00:40:00
there.
So, how to do max/min problems
531
00:40:00 --> 00:40:08
with non-independent variables
-- So, if your variables are
532
00:40:08 --> 00:40:15
related by some condition,
g equals some constant.
533
00:40:15 --> 00:40:25
So, then we've seen the method
of Lagrange multipliers.
534
00:40:25 --> 00:40:31
OK, and what this method says
is that we should solve the
535
00:40:31 --> 00:40:36
equation gradient f equals some
unknown scalar lambda times the
536
00:40:36 --> 00:40:39
gradient, g.
So, that means each partial,
537
00:40:39 --> 00:40:43
f sub x equals lambda g sub x
and so on,
538
00:40:43 --> 00:40:48
and of course we have to keep
in mind the constraint equation
539
00:40:48 --> 00:40:53
so that we have the same number
of equations as the number of
540
00:40:53 --> 00:40:57
unknowns because you have a new
unknown here.
541
00:40:57 --> 00:41:04
And, the thing to remember is
that you have to be careful that
542
00:41:04 --> 00:41:13
the second derivative test does
not apply in this situation.
543
00:41:13 --> 00:41:16
I mean, this is only in the
case of independent variables.
544
00:41:16 --> 00:41:18
So, if you want to know if
something is a maximum or a
545
00:41:18 --> 00:41:20
minimum,
you just have to use common
546
00:41:20 --> 00:41:24
sense or compare the values of a
function at the various points
547
00:41:24 --> 00:41:29
you found.
Yes?
548
00:41:29 --> 00:41:34
Will we actually have to
calculate?
549
00:41:34 --> 00:41:38
Well, that depends on what the
problem asks you.
550
00:41:38 --> 00:41:40
It might ask you to just set up
the equations,
551
00:41:40 --> 00:41:41
or it might ask you to solve
them.
552
00:41:41 --> 00:41:44
So, in general,
solving might be difficult,
553
00:41:44 --> 00:41:47
but if it asks you to do it,
then it means it shouldn't be
554
00:41:47 --> 00:41:50
too hard.
I haven't written the final
555
00:41:50 --> 00:41:54
yet, so I don't know what it
will be, but it might be an easy
556
00:41:54 --> 00:42:00
one.
And, the last thing we've seen
557
00:42:00 --> 00:42:06
is constrained partial
derivatives.
558
00:42:06 --> 00:42:12
So, for example,
if you have a relation between
559
00:42:12 --> 00:42:15
x, y, and z,
which are constrained to be a
560
00:42:15 --> 00:42:20
constant,
then the notion of partial f
561
00:42:20 --> 00:42:24
partial x takes several
meanings.
562
00:42:24 --> 00:42:32
So, just to remind you very
quickly, there's the formal
563
00:42:32 --> 00:42:38
partial, partial f,
partial x, which means x
564
00:42:38 --> 00:42:43
varies.
Y and z are held constant.
565
00:42:43 --> 00:42:48
And, we forget the constraint.
This is not compatible with a
566
00:42:48 --> 00:42:51
constraint, but we don't care.
So, that's the guy that we
567
00:42:51 --> 00:42:54
compute just from the formula
for f ignoring the constraints.
568
00:42:54 --> 00:43:01
And then, we have the partial
f, partial x with y held
569
00:43:01 --> 00:43:06
constant, which means y held
constant.
570
00:43:06 --> 00:43:15
X varies, and now we treat z as
a dependent variable.
571
00:43:15 --> 00:43:20
It varies with x and y
according to whatever is needed
572
00:43:20 --> 00:43:24
so that this constraint keeps
holding.
573
00:43:24 --> 00:43:29
And, similarly,
there's partial f partial x
574
00:43:29 --> 00:43:33
with z held constant,
which means that,
575
00:43:33 --> 00:43:38
now, y is the dependent
variable.
576
00:43:38 --> 00:43:39
And, the way in which we
compute these,
577
00:43:39 --> 00:43:42
we've seen two methods which
I'm not going to tell you now
578
00:43:42 --> 00:43:45
because otherwise we'll be even
more over time.
579
00:43:45 --> 00:43:48
But, we've seen two methods for
computing these based on either
580
00:43:48 --> 00:43:50
the chain rule or on
differentials,
581
00:43:50 --> 00:43:52
solving and substituting into
differentials.
582
00:43:52 --> 00:43:53