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OK, so anyway,
let's get started.
00:00:27.000 --> 00:00:31.000
So,
the first unit of the class,
00:00:31.000 --> 00:00:33.000
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
00:00:36.000 --> 00:00:41.000
class on Tuesday just because we
have to start somewhere.
00:00:41.000 --> 00:00:48.000
So, the first things that we
learned about in this class were
00:00:48.000 --> 00:00:54.000
vectors, and how to do
dot-product of vectors.
00:00:54.000 --> 00:01:01.000
So, remember the formula that A
dot B is the sum of ai times bi.
00:01:01.000 --> 00:01:05.000
And, geometrically,
it's length A times length B
00:01:05.000 --> 00:01:08.000
times the cosine of the angle
between them.
00:01:08.000 --> 00:01:11.000
And, in particular,
we can use this to detect when
00:01:11.000 --> 00:01:14.000
two vectors are perpendicular.
That's when their dot product
00:01:14.000 --> 00:01:17.000
is zero.
And, we can use that to measure
00:01:17.000 --> 00:01:21.000
angles between vectors by
solving for cosine in this.
00:01:21.000 --> 00:01:25.000
Hopefully, at this point,
this looks a lot easier than it
00:01:25.000 --> 00:01:28.000
used to a few months ago.
So, hopefully at this point,
00:01:28.000 --> 00:01:32.000
everyone has this kind of
formula memorized and has some
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reasonable understanding of
that.
00:01:35.000 --> 00:01:41.000
But, if you have any questions,
now is the time.
00:01:41.000 --> 00:01:45.000
No?
Good.
00:01:45.000 --> 00:01:55.000
Next we learned how to also do
cross product of vectors in
00:01:55.000 --> 00:02:06.000
space -- -- and remember,
we saw how to use that to find
00:02:06.000 --> 00:02:10.000
area of, say,
a triangle or a parallelogram
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in space because the length of
the cross product is equal to
00:02:14.000 --> 00:02:17.000
the area of a parallelogram
formed by the vectors a and b.
00:02:17.000 --> 00:02:25.000
And, we can also use that to
find a vector perpendicular to
00:02:25.000 --> 00:02:28.000
two given vectors,
A and B.
00:02:28.000 --> 00:02:33.000
And so, in particular,
that comes in handy when we are
00:02:33.000 --> 00:02:42.000
looking for the equation of a
plane because we've seen -- So,
00:02:42.000 --> 00:02:49.000
the next topic would be
equations of planes.
00:02:49.000 --> 00:02:55.000
And, we've seen that when you
put the equation of a plane in
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the form ax by cz = d,
well, 00:03:03.000
b, c> in there is actually
the normal vector to the plane,
00:03:03.000 --> 00:03:07.000
or some normal vector to the
plane.
00:03:07.000 --> 00:03:11.000
So, typically,
we use cross product to find
00:03:11.000 --> 00:03:16.000
plane equations.
OK, is that still reasonably
00:03:16.000 --> 00:03:21.000
familiar to everyone?
Yes, very good.
00:03:21.000 --> 00:03:26.000
OK, we've also seen how to look
at equations of lines,
00:03:26.000 --> 00:03:31.000
and those were of a slightly
different nature because we've
00:03:31.000 --> 00:03:35.000
been doing them as parametric
equations.
00:03:35.000 --> 00:03:42.000
So, typically we had equations
of a form, maybe x equals some
00:03:42.000 --> 00:03:47.000
constant times t,
y equals constant plus constant
00:03:47.000 --> 00:03:53.000
times t.
z equals constant plus constant
00:03:53.000 --> 00:04:02.000
times t where these terms here
correspond to some point on the
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line.
And, these coefficients here
00:04:06.000 --> 00:04:11.000
correspond to a vector parallel
to the line.
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That's the velocity of the
moving point on the line.
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And, well,
we've learned in particular how
00:04:23.000 --> 00:04:29.000
to find where a line intersects
a plane by plugging in the
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parametric equation into the
equation of a plane.
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We've learned more general
things about parametric
00:04:43.000 --> 00:04:48.000
equations of curves.
So, there are these infamous
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problems in particular where you
have these rotating wheels and
00:04:51.000 --> 00:04:53.000
points on them,
and you have to figure out,
00:04:53.000 --> 00:04:57.000
what's the position of a point?
And, the general principle of
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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
00:05:05.000 --> 00:05:08.000
wheel that's itself moving and
something else,
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then you might want to first
figure out the position of a
00:05:11.000 --> 00:05:14.000
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
00:05:18.000 --> 00:05:23.000
a moving point by adding
together simpler vectors.
00:05:23.000 --> 00:05:27.000
So, the general principle is
really to try to find one
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parameter that will let us
understand what has happened,
00:05:30.000 --> 00:05:36.000
and then decompose the motion
into a sum of simpler effect.
00:05:36.000 --> 00:05:54.000
So, we want to decompose the
position vector into a sum of
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simpler vectors.
OK, so maybe now we are getting
00:06:02.000 --> 00:06:05.000
a bit out of some people's
comfort zone,
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but hopefully it's not too bad.
Do you have any general
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questions about how one would go
about that, or,
00:06:20.000 --> 00:06:24.000
yes?
Sorry? What about it?
00:06:24.000 --> 00:06:25.000
Parametric descriptions of a
plane,
00:06:25.000 --> 00:06:28.000
so we haven't really done that
because you would need two
00:06:28.000 --> 00:06:31.000
parameters to parameterize a
plane just because it's a two
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dimensional object.
So, we have mostly focused on
00:06:35.000 --> 00:06:40.000
the use of parametric equations
just for one dimensional
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objects, lines,
and curves.
00:06:42.000 --> 00:06:45.000
So,
you won't need to know about
00:06:45.000 --> 00:06:47.000
parametric descriptions of
planes on a final,
00:06:47.000 --> 00:06:51.000
but if you really wanted to,
you would think of defining a
00:06:51.000 --> 00:06:55.000
point on a plane as starting
from some given point.
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Then you have two vectors given
on the plane.
00:06:57.000 --> 00:07:00.000
And then, you would add a
multiple of each of these
00:07:00.000 --> 00:07:04.000
vectors to your starting point.
But see, the difficulty is to
00:07:04.000 --> 00:07:08.000
convert from that to the usual
equation of a plane,
00:07:08.000 --> 00:07:11.000
you would still have to go back
to this cross product method,
00:07:11.000 --> 00:07:15.000
and so on.
So, it is possible to represent
00:07:15.000 --> 00:07:19.000
a plane, or, in general,
a surface in parametric form.
00:07:19.000 --> 00:07:23.000
But, very often,
that's not so useful.
00:07:23.000 --> 00:07:28.000
Yes?
How do you parametrize an
00:07:28.000 --> 00:07:31.000
ellipse in space?
Well, that depends on how it's
00:07:31.000 --> 00:07:34.000
given to you.
But, OK, let's just do an
00:07:34.000 --> 00:07:38.000
example.
Say that I give you an ellipse
00:07:38.000 --> 00:07:42.000
in space as maybe the more,
well, one exciting way to
00:07:42.000 --> 00:07:45.000
parameterize an ellipse in space
is maybe the intersection of a
00:07:45.000 --> 00:07:49.000
cylinder with a slanted plane.
That's the kind of situations
00:07:49.000 --> 00:07:52.000
where you might end up with an
ellipse.
00:07:52.000 --> 00:07:58.000
OK, so if I tell you that maybe
I'm intersecting a cylinder with
00:07:58.000 --> 00:08:03.000
equation x squared plus y
squared equals a squared with a
00:08:03.000 --> 00:08:09.000
slanted plane to get,
I messed up my picture,
00:08:09.000 --> 00:08:13.000
to get this ellipse of
intersection,
00:08:13.000 --> 00:08:14.000
so, of course you'd need the
equation of a plane.
00:08:14.000 --> 00:08:18.000
And, let's say that this plane
is maybe given to you.
00:08:18.000 --> 00:08:23.000
Or, you can switch it to form
where you can get z as a
00:08:23.000 --> 00:08:29.000
function of x and y.
So, maybe it would be z equals,
00:08:29.000 --> 00:08:33.000
I've already used a;
I need to use a new letter.
00:08:33.000 --> 00:08:41.000
Let's say c1x c2y plus d,
whatever, something like that.
00:08:41.000 --> 00:08:45.000
So, what I would do is first I
would look at what my ellipse
00:08:45.000 --> 00:08:49.000
does in the directions in which
I understand it the best.
00:08:49.000 --> 00:08:53.000
And, those directions would be
probably the xy plane.
00:08:53.000 --> 00:08:56.000
So, I would look at the xy
coordinates.
00:08:56.000 --> 00:09:02.000
Well, if I look at it from
above xy, my ellipse looks like
00:09:02.000 --> 00:09:06.000
just a circle of radius a.
So, if I'm only concerned with
00:09:06.000 --> 00:09:10.000
x and y, presumably I can just
do it the usual way for a
00:09:10.000 --> 00:09:13.000
circle.
x equals a cosine t.
00:09:13.000 --> 00:09:20.000
y equals a sine t, OK?
And then, z would end up being
00:09:20.000 --> 00:09:24.000
just, well, whatever the value
of z is to be on the slanted
00:09:24.000 --> 00:09:29.000
plane above a given xy position.
So, in fact,
00:09:29.000 --> 00:09:38.000
it would end up being ac1
cosine t plus ac2 sine t plus d,
00:09:38.000 --> 00:09:42.000
I guess.
OK, that's not a particularly
00:09:42.000 --> 00:09:44.000
elegant parameterization,
but that's the kind of thing
00:09:44.000 --> 00:09:47.000
you might end up with.
Now, in general,
00:09:47.000 --> 00:09:50.000
when you have a curve in space,
it would rarely be the case
00:09:50.000 --> 00:09:53.000
that you have to get a
parameterization from scratch
00:09:53.000 --> 00:09:56.000
unless you are already being
told information about how it
00:09:56.000 --> 00:09:58.000
looks in one of the coordinate
planes,
00:09:58.000 --> 00:10:03.000
this kind of method.
Or, at least you'd have a lot
00:10:03.000 --> 00:10:07.000
of information that would
quickly reduce to a plane
00:10:07.000 --> 00:10:11.000
problem somehow.
Of course, I could also just
00:10:11.000 --> 00:10:16.000
give you some formulas and let
you figure out what's going on.
00:10:16.000 --> 00:10:21.000
But, in general,
we've done more stuff with
00:10:21.000 --> 00:10:25.000
plane curves.
With plane curves,
00:10:25.000 --> 00:10:29.000
certainly there's interesting
things with all sorts of
00:10:29.000 --> 00:10:32.000
mechanical gadgets that we can
study.
00:10:32.000 --> 00:10:39.000
OK, any other questions on that?
No?
00:10:39.000 --> 00:10:45.000
OK, so let me move on a bit and
point out that with parametric
00:10:45.000 --> 00:10:51.000
equations, we've looked also at
things like velocity and
00:10:51.000 --> 00:10:55.000
acceleration.
So, the velocity vector is the
00:10:55.000 --> 00:10:59.000
derivative of a position vector
with respect to time.
00:10:59.000 --> 00:11:04.000
And, it's not to be confused
with speed, which is the
00:11:04.000 --> 00:11:08.000
magnitude of v.
So, the velocity vector is
00:11:08.000 --> 00:11:12.000
going to be always tangent to
the curve.
00:11:12.000 --> 00:11:14.000
And, its length will be the
speed.
00:11:14.000 --> 00:11:15.000
That's the geometric
interpretation.
00:11:32.000 --> 00:11:37.000
So, just to provoke you,
I'm going to write,
00:11:37.000 --> 00:11:43.000
again, that formula that was
that v equals T hat ds dt.
00:11:43.000 --> 00:11:46.000
What do I mean by that?
If I have a curve,
00:11:46.000 --> 00:11:51.000
and I'm moving on the curve,
well, I have the unit tangent
00:11:51.000 --> 00:11:56.000
vector which I think at the time
I used to draw in blue.
00:11:56.000 --> 00:11:59.000
But, blue has been abolished
since then.
00:11:59.000 --> 00:12:04.000
So, I'm going to draw it in red.
OK, so that's a unit vector
00:12:04.000 --> 00:12:09.000
that goes along the curve,
and then the actual velocity is
00:12:09.000 --> 00:12:11.000
going to be proportional to
that.
00:12:11.000 --> 00:12:15.000
And, what's the length?
Well, it's the speed.
00:12:15.000 --> 00:12:19.000
And, the speed is how much arc
length on the curve I go per
00:12:19.000 --> 00:12:22.000
unit time, which is why I'm
writing ds dt.
00:12:22.000 --> 00:12:30.000
That's another guy.
That's another of these guys
00:12:30.000 --> 00:12:34.000
for the speed,
OK?
00:12:34.000 --> 00:12:41.000
And, we've also learned about
acceleration,
00:12:41.000 --> 00:12:47.000
which is the derivative of
velocity.
00:12:47.000 --> 00:12:50.000
So, it's the second derivative
of a position vector.
00:12:50.000 --> 00:12:54.000
And, as an example of the kinds
of manipulations we can do,
00:12:54.000 --> 00:12:56.000
in class we've seen Kepler's
second law,
00:12:56.000 --> 00:13:03.000
which explains how if the
acceleration is parallel to the
00:13:03.000 --> 00:13:08.000
position vector,
then r cross v is going to be
00:13:08.000 --> 00:13:10.000
constant,
which means that the motion
00:13:10.000 --> 00:13:13.000
will be in an plane,
and you will sweep area at a
00:13:13.000 --> 00:13:16.000
constant rate.
So now, that is not in itself a
00:13:16.000 --> 00:13:19.000
topic for the exam,
but the kinds of methods of
00:13:19.000 --> 00:13:22.000
differentiating vector
quantities,
00:13:22.000 --> 00:13:25.000
applying the product rule to
take the derivative of a dot or
00:13:25.000 --> 00:13:28.000
cross product and so on are
definitely fair game.
00:13:28.000 --> 00:13:30.000
I mean, we've seen those on the
first exam.
00:13:30.000 --> 00:13:35.000
They were there,
and most likely they will be on
00:13:35.000 --> 00:13:39.000
the final.
OK, so I mean that's the extent
00:13:39.000 --> 00:13:44.000
to which Kepler's law comes up,
only just knowing the general
00:13:44.000 --> 00:13:47.000
type of manipulations and
proving things with vector
00:13:47.000 --> 00:13:52.000
quantities,
but not again the actual
00:13:52.000 --> 00:13:58.000
Kepler's law itself.
I skipped something.
00:13:58.000 --> 00:14:08.000
I skipped matrices,
determinants,
00:14:08.000 --> 00:14:18.000
and linear systems.
OK, so we've seen how to
00:14:18.000 --> 00:14:24.000
multiply matrices,
and how to write linear systems
00:14:24.000 --> 00:14:28.000
in matrix form.
So, remember,
00:14:28.000 --> 00:14:35.000
if you have a 3x3 linear system
in the usual sense,
00:14:35.000 --> 00:14:42.000
so,
you can write this in a matrix
00:14:42.000 --> 00:14:52.000
form where you have a 3x3 matrix
and you have an unknown column
00:14:52.000 --> 00:14:57.000
vector.
And, their matrix product
00:14:57.000 --> 00:15:01.000
should be some given column
vector.
00:15:01.000 --> 00:15:04.000
OK, so if you don't remember
how to multiply matrices,
00:15:04.000 --> 00:15:07.000
please look at the notes on
that again.
00:15:07.000 --> 00:15:12.000
And, also you should remember
how to invert a matrix.
00:15:12.000 --> 00:15:22.000
So, how did we invert matrices?
Let me just remind you very
00:15:22.000 --> 00:15:30.000
quickly.
So, I should say 2x2 or 3x3
00:15:30.000 --> 00:15:33.000
matrices.
Well, you need to have a square
00:15:33.000 --> 00:15:35.000
matrix to be able to find an
inverse.
00:15:35.000 --> 00:15:37.000
The method doesn't work,
doesn't make sense.
00:15:37.000 --> 00:15:40.000
Otherwise, then the concept of
inverse doesn't work.
00:15:40.000 --> 00:15:43.000
And, if it's larger than 3x3,
then we haven't seen that.
00:15:43.000 --> 00:15:50.000
So, let's say that I have a 3x3
matrix.
00:15:50.000 --> 00:16:00.000
What I will do is I will start
by forming the matrix of minors.
00:16:00.000 --> 00:16:09.000
So, remember that minors,
so, each entry is a 2x2
00:16:09.000 --> 00:16:20.000
determinant in the case of a 3x3
matrix formed by deleting one
00:16:20.000 --> 00:16:26.000
row and one column.
OK, so for example,
00:16:26.000 --> 00:16:30.000
to get the first minor,
especially in the upper left
00:16:30.000 --> 00:16:34.000
corner, I would delete the first
row, the first column.
00:16:34.000 --> 00:16:36.000
And, I would be left with this
2x2 determinant.
00:16:36.000 --> 00:16:38.000
I take this times that minus
this times that.
00:16:38.000 --> 00:16:41.000
I get a number that gives my
first minor.
00:16:41.000 --> 00:16:49.000
And then, same with the others.
Then, I flip signs according to
00:16:49.000 --> 00:16:56.000
this checkerboard pattern,
and that gives me the matrix of
00:16:56.000 --> 00:17:00.000
cofactors.
OK, so all it means is I'm just
00:17:00.000 --> 00:17:06.000
changing the signs of these four
entries and leaving the others
00:17:06.000 --> 00:17:10.000
alone.
And then, I take the transpose
00:17:10.000 --> 00:17:13.000
of that.
So, that means I read it
00:17:13.000 --> 00:17:16.000
horizontally and write it down
vertically.
00:17:16.000 --> 00:17:19.000
I swept the rows and the
columns.
00:17:19.000 --> 00:17:23.000
And then, I divide by the
inverse.
00:17:23.000 --> 00:17:28.000
Well, I divide by the
determinant of the initial
00:17:28.000 --> 00:17:30.000
matrix.
OK, so, of course,
00:17:30.000 --> 00:17:32.000
this is kind of very
theoretical, and I write it like
00:17:32.000 --> 00:17:34.000
this.
Probably it makes more sense to
00:17:34.000 --> 00:17:37.000
do it on an example.
I will let you work out
00:17:37.000 --> 00:17:42.000
examples, or bug your recitation
instructors so that they do one
00:17:42.000 --> 00:17:44.000
on Monday if you want to see
that.
00:17:44.000 --> 00:17:47.000
It's a fairly straightforward
method.
00:17:47.000 --> 00:17:50.000
You just have to remember the
steps.
00:17:50.000 --> 00:17:52.000
But, of course,
there's one condition,
00:17:52.000 --> 00:17:57.000
which is that the determinant
of a matrix has to be nonzero.
00:17:57.000 --> 00:17:59.000
So, in fact,
we've seen that,
00:17:59.000 --> 00:18:03.000
oh, there is still one board
left.
00:18:03.000 --> 00:18:12.000
We've seen that a matrix is
invertible -- -- exactly when
00:18:12.000 --> 00:18:19.000
its determinant is not zero.
And, if that's the case,
00:18:19.000 --> 00:18:24.000
then we can solve the linear
system, AX equals B by just
00:18:24.000 --> 00:18:30.000
setting X equals A inverse B.
That's going to be the only
00:18:30.000 --> 00:18:38.000
solution to our linear system.
Otherwise, well,
00:18:38.000 --> 00:18:52.000
AX equals B has either no
solution, or infinitely many
00:18:52.000 --> 00:19:01.000
solutions.
Yes?
00:19:01.000 --> 00:19:04.000
The determinant of a matrix
real quick?
00:19:04.000 --> 00:19:08.000
Well, I can do it that quickly
unless I start waving my hands
00:19:08.000 --> 00:19:12.000
very quickly,
but remember we've seen that
00:19:12.000 --> 00:19:15.000
you have a matrix,
a 3x3 matrix.
00:19:15.000 --> 00:19:18.000
Its determinant will be
obtained by doing an expansion
00:19:18.000 --> 00:19:20.000
with respect to,
well, your favorite.
00:19:20.000 --> 00:19:22.000
But usually,
we are doing it with respect to
00:19:22.000 --> 00:19:26.000
the first row.
So, we take this entry and
00:19:26.000 --> 00:19:31.000
multiply it by that determinant.
Then, we take that entry,
00:19:31.000 --> 00:19:35.000
multiply it by that determinant
but put a minus sign.
00:19:35.000 --> 00:19:38.000
And then, we take that entry
and multiply it by this
00:19:38.000 --> 00:19:41.000
determinant here,
and we put a plus sign for
00:19:41.000 --> 00:19:44.000
that.
OK, so maybe I should write it
00:19:44.000 --> 00:19:46.000
down.
That's actually the same
00:19:46.000 --> 00:19:48.000
formula that we are using for
cross products.
00:19:48.000 --> 00:19:50.000
Right, when we do cross
products, we are doing an
00:19:50.000 --> 00:19:53.000
expansion with respect to the
first row.
00:19:53.000 --> 00:19:57.000
That's a special case.
OK, I mean, do you still want
00:19:57.000 --> 00:19:59.000
to see it in more details,
or is that OK?
00:19:59.000 --> 00:20:12.000
Yes?
That's correct.
00:20:12.000 --> 00:20:16.000
So, if you do an expansion with
respect to any row or column,
00:20:16.000 --> 00:20:19.000
then you would use the same
signs that are in this
00:20:19.000 --> 00:20:22.000
checkerboard pattern there.
So, if you did an expansion,
00:20:22.000 --> 00:20:25.000
actually, so indeed,
maybe I should say,
00:20:25.000 --> 00:20:28.000
the more general way to
determine it is you take your
00:20:28.000 --> 00:20:31.000
favorite row or column,
and you just multiply the
00:20:31.000 --> 00:20:34.000
corresponding entries by the
corresponding cofactors.
00:20:34.000 --> 00:20:37.000
So, the signs are plus or minus
depending on what's in that
00:20:37.000 --> 00:20:38.000
diagram there.
Now, in practice,
00:20:38.000 --> 00:20:41.000
in this class,
again, all we need is to do it
00:20:41.000 --> 00:20:46.000
with respect to the first row.
So, don't worry about it too
00:20:46.000 --> 00:20:48.000
much.
OK, so, again,
00:20:48.000 --> 00:20:51.000
the way that we've officially
seen it in this class is just if
00:20:51.000 --> 00:20:59.000
you have a1,
a2, a3, b1, b2, b3, c1, c2, c3,
00:20:59.000 --> 00:21:06.000
so if the determinant is a1
times b2 b3, c2 c3,
00:21:06.000 --> 00:21:16.000
minus a2 b1 b3 c1 c3 plus a3 b1
b2 c1 c2.
00:21:16.000 --> 00:21:20.000
And, this minus is here
basically because of the minus
00:21:20.000 --> 00:21:27.000
in the diagram up there.
But, that's all we need to know.
00:21:27.000 --> 00:21:32.000
Yes?
How do you tell the difference
00:21:32.000 --> 00:21:34.000
between infinitely many
solutions or no solutions?
00:21:34.000 --> 00:21:37.000
That's a very good question.
So, in full generality,
00:21:37.000 --> 00:21:40.000
the answer is we haven't quite
seen a systematic method.
00:21:40.000 --> 00:21:43.000
So, you just have to try
solving and see if you can find
00:21:43.000 --> 00:21:46.000
a solution or not.
So, let me actually explain
00:21:46.000 --> 00:21:51.000
that more carefully.
So, what happens to these two
00:21:51.000 --> 00:21:56.000
situations when a is invertible
or not?
00:21:56.000 --> 00:21:57.000
So, remember,
in the linear system,
00:21:57.000 --> 00:22:01.000
you can think of a linear
system as asking you to find the
00:22:01.000 --> 00:22:05.000
intersection between three
planes because each equation is
00:22:05.000 --> 00:22:12.000
the equation of a plane.
So, Ax = B for a 3x3 system
00:22:12.000 --> 00:22:24.000
means that x should be in the
intersection of three planes.
00:22:24.000 --> 00:22:28.000
And then, we have two cases.
So, the case where the system
00:22:28.000 --> 00:22:33.000
is invertible corresponds to the
general situation where your
00:22:33.000 --> 00:22:37.000
three planes somehow all just
intersect in one point.
00:22:37.000 --> 00:22:41.000
And then, the situation where
the determinant,
00:22:41.000 --> 00:22:45.000
that's when the determinant is
not zero, you get just one
00:22:45.000 --> 00:22:48.000
point.
However, sometimes it will
00:22:48.000 --> 00:22:54.000
happen that all the planes are
parallel to the same direction.
00:22:54.000 --> 00:23:04.000
So, determinant a equals zero
means the three planes are
00:23:04.000 --> 00:23:11.000
parallel to a same vector.
And, in fact,
00:23:11.000 --> 00:23:14.000
you can find that vector
explicitly because that vector
00:23:14.000 --> 00:23:17.000
has to be perpendicular to all
the normals.
00:23:17.000 --> 00:23:22.000
So, at some point we saw other
subtle things about how to find
00:23:22.000 --> 00:23:26.000
the direction of this line
that's parallel to all the
00:23:26.000 --> 00:23:30.000
planes.
So, now, this can happen either
00:23:30.000 --> 00:23:34.000
with all three planes containing
the same line.
00:23:34.000 --> 00:23:36.000
You know, they can all pass
through the same axis.
00:23:36.000 --> 00:23:39.000
Or it could be that they have
somehow shifted with respect to
00:23:39.000 --> 00:23:44.000
each other.
And so, it might look like this.
00:23:44.000 --> 00:23:46.000
Then, the last one is actually
in front of that.
00:23:46.000 --> 00:23:52.000
So, see, the lines of
intersections between two of the
00:23:52.000 --> 00:23:55.000
planes,
so, here they all pass through
00:23:55.000 --> 00:23:57.000
the same line,
and here, instead,
00:23:57.000 --> 00:24:00.000
they intersect in one line
here,
00:24:00.000 --> 00:24:03.000
one line here,
and one line there.
00:24:03.000 --> 00:24:06.000
And, there's no triple
intersection.
00:24:06.000 --> 00:24:08.000
So, in general,
we haven't really seen how to
00:24:08.000 --> 00:24:13.000
decide between these two cases.
There's one important situation
00:24:13.000 --> 00:24:20.000
where we have seen we must be in
the first case that when we have
00:24:20.000 --> 00:24:26.000
a homogeneous system,
so that means if the right hand
00:24:26.000 --> 00:24:31.000
side is zero,
then,
00:24:31.000 --> 00:24:41.000
well, x equals zero is always a
solution.
00:24:41.000 --> 00:24:43.000
It's called the trivial
solution.
00:24:43.000 --> 00:24:50.000
It's the obvious one,
if you want.
00:24:50.000 --> 00:24:53.000
So, you know that,
and why is that?
00:24:53.000 --> 00:24:57.000
Well, that's because all of
your planes have to pass through
00:24:57.000 --> 00:25:00.000
the origin.
So, you must be in this case if
00:25:00.000 --> 00:25:04.000
you have a noninvertible system
where the right hand side is
00:25:04.000 --> 00:25:05.000
zero.
So, in that case,
00:25:05.000 --> 00:25:08.000
if the right hand side is zero,
there's two cases.
00:25:08.000 --> 00:25:12.000
Either the matrix is invertible.
Then, the only solution is the
00:25:12.000 --> 00:25:14.000
trivial one.
Or, if a matrix is not
00:25:14.000 --> 00:25:19.000
invertible, then you have
infinitely many solutions.
00:25:19.000 --> 00:25:23.000
If B is not zero,
then we haven't really seen how
00:25:23.000 --> 00:25:27.000
to decide.
We've just seen how to decide
00:25:27.000 --> 00:25:30.000
between one solution or
zero,infinitely many,
00:25:30.000 --> 00:25:33.000
but not how to decide between
these last two cases.
00:25:33.000 --> 00:25:42.000
Yes?
I think in principle,
00:25:42.000 --> 00:25:44.000
you would be able to,
but that's, well,
00:25:44.000 --> 00:25:48.000
I mean, that's a slightly
counterintuitive way of doing
00:25:48.000 --> 00:25:50.000
it.
I think it would probably work.
00:25:50.000 --> 00:25:55.000
Well, I'll let you figure it
out.
00:25:55.000 --> 00:25:59.000
OK, let me move on to the
second unit, maybe,
00:25:59.000 --> 00:26:03.000
because we've seen a lot of
stuff, or was there a quick
00:26:03.000 --> 00:26:05.000
question before that?
OK.
00:26:41.000 --> 00:26:44.000
OK, so what was the second part
of the class about?
00:26:44.000 --> 00:26:47.000
Well, hopefully you kind of
vaguely remember that it was
00:26:47.000 --> 00:26:50.000
about functions of several
variables and their partial
00:26:50.000 --> 00:26:55.000
derivatives.
OK, so the first thing that
00:26:55.000 --> 00:27:04.000
we've seen is how to actually
view a function of two variables
00:27:04.000 --> 00:27:12.000
in terms of its graph and its
contour plot.
00:27:12.000 --> 00:27:15.000
So,
just to remind you very
00:27:15.000 --> 00:27:17.000
quickly,
if I have a function of two
00:27:17.000 --> 00:27:21.000
variables, x and y,
then the graph will be just the
00:27:21.000 --> 00:27:25.000
surface given by the equation z
equals f of xy.
00:27:25.000 --> 00:27:28.000
So, for each x and y,
I plot a point at height given
00:27:28.000 --> 00:27:30.000
with the value of the a
function.
00:27:30.000 --> 00:27:34.000
And then, the contour plot will
be the topographical map for
00:27:34.000 --> 00:27:37.000
this graph.
It will tell us,
00:27:37.000 --> 00:27:41.000
what are the various levels in
there?
00:27:41.000 --> 00:27:46.000
So, what it amounts to is we
slice the graph by horizontal
00:27:46.000 --> 00:27:50.000
planes, and we get a bunch of
curves which are the points at
00:27:50.000 --> 00:27:56.000
given height on the plot.
And, so we get all of these
00:27:56.000 --> 00:28:04.000
curves, and then we look at them
from above, and that gives us
00:28:04.000 --> 00:28:09.000
this map with a bunch of curves
on it.
00:28:09.000 --> 00:28:13.000
And, each of them has a number
next to it which tells us the
00:28:13.000 --> 00:28:16.000
value of a function there.
And, from that map, we can,
00:28:16.000 --> 00:28:19.000
of course, tell things about
where we might be able to find
00:28:19.000 --> 00:28:22.000
minima or maxima of our
function,
00:28:22.000 --> 00:28:30.000
and how it varies with respect
to x or y or actually in any
00:28:30.000 --> 00:28:40.000
direction at a given point.
So, now, the next thing that
00:28:40.000 --> 00:28:49.000
we've learned about is partial
derivatives.
00:28:49.000 --> 00:28:52.000
So, for a function of two
variables, there would be two of
00:28:52.000 --> 00:28:54.000
them.
There's f sub x which is
00:28:54.000 --> 00:28:58.000
partial f partial x,
and f sub y which is partial f
00:28:58.000 --> 00:29:00.000
partial y.
And, in terms of a graph,
00:29:00.000 --> 00:29:04.000
they correspond to slicing by a
plane that's parallel to one of
00:29:04.000 --> 00:29:07.000
the coordinate planes,
so that we either keep x
00:29:07.000 --> 00:29:10.000
constant,
or keep y constant.
00:29:10.000 --> 00:29:14.000
And, we look at the slope of a
graph to see the rate of change
00:29:14.000 --> 00:29:17.000
of f with respect to one
variable only when we hold the
00:29:17.000 --> 00:29:21.000
other one constant.
And so, we've seen in
00:29:21.000 --> 00:29:25.000
particular how to use that in
various places,
00:29:25.000 --> 00:29:29.000
but, for example,
for linear approximation we've
00:29:29.000 --> 00:29:34.000
seen that the change in f is
approximately equal to f sub x
00:29:34.000 --> 00:29:40.000
times the change in x plus f sub
y times the change in y.
00:29:40.000 --> 00:29:45.000
So, you can think of f sub x
and f sub y as telling you how
00:29:45.000 --> 00:29:49.000
sensitive the value of f is to
changes in x and y.
00:29:49.000 --> 00:29:59.000
So, this linear approximation
also tells us about the tangent
00:29:59.000 --> 00:30:07.000
plane to the graph of f.
In fact, when we turn this into
00:30:07.000 --> 00:30:16.000
an equality, that would mean
that we replace f by the tangent
00:30:16.000 --> 00:30:19.000
plane.
We've also learned various ways
00:30:19.000 --> 00:30:21.000
of, before I go on,
I should say,
00:30:21.000 --> 00:30:24.000
of course, we've seen these
also for functions of three
00:30:24.000 --> 00:30:28.000
variables, right?
So, we haven't seen how to plot
00:30:28.000 --> 00:30:32.000
them, and we don't really worry
about that too much.
00:30:32.000 --> 00:30:37.000
But, if you have a function of
three variables,
00:30:37.000 --> 00:30:42.000
you can do the same kinds of
manipulations.
00:30:42.000 --> 00:30:49.000
So, we've learned about
differentials and chain rules,
00:30:49.000 --> 00:30:57.000
which are a way of repackaging
these partial derivatives.
00:30:57.000 --> 00:31:00.000
So, the differential is just,
by definition,
00:31:00.000 --> 00:31:05.000
this thing called df which is f
sub x times dx plus f sub y
00:31:05.000 --> 00:31:09.000
times dy.
And, what we can do with it is
00:31:09.000 --> 00:31:14.000
just either plug values for
changes in x and y,
00:31:14.000 --> 00:31:17.000
and get approximation formulas,
or we can look at this in a
00:31:17.000 --> 00:31:21.000
situation where x and y will
depend on something else,
00:31:21.000 --> 00:31:26.000
and we get a chain rule.
So, for example,
00:31:26.000 --> 00:31:32.000
if f is a function of t time,
for example, and so is y,
00:31:32.000 --> 00:31:36.000
then we can find the rate of
change of f with respect to t
00:31:36.000 --> 00:31:43.000
just by dividing this by dt.
So, we get df dt equals f sub x
00:31:43.000 --> 00:31:48.000
dx dt plus f sub y dy dt.
We can also get other chain
00:31:48.000 --> 00:31:51.000
rules,
say, if x and y depend on more
00:31:51.000 --> 00:31:54.000
than one variable,
if you have a change of
00:31:54.000 --> 00:31:55.000
variables,
for example,
00:31:55.000 --> 00:31:58.000
x and y are functions of two
other guys that you call u and
00:31:58.000 --> 00:32:01.000
v,
then you can express dx and dy
00:32:01.000 --> 00:32:05.000
in terms of du and dv,
and plugging into df you will
00:32:05.000 --> 00:32:08.000
get the manner in which f
depends on u and v.
00:32:08.000 --> 00:32:11.000
So, that will give you formulas
for partial f partial u,
00:32:11.000 --> 00:32:14.000
and partial f partial v.
They look just like these guys
00:32:14.000 --> 00:32:19.000
except there's a lot of curly
d's instead of straight ones,
00:32:19.000 --> 00:32:21.000
and u's and v's in the
denominators.
00:32:21.000 --> 00:32:26.000
OK, so that lets us understand
rates of change.
00:32:26.000 --> 00:32:31.000
We've also seen yet another way
to package partial derivatives
00:32:31.000 --> 00:32:33.000
into not a differential,
but instead,
00:32:33.000 --> 00:32:37.000
a vector.
That's the gradient vector,
00:32:37.000 --> 00:32:41.000
and I'm sure it was quite
mysterious when we first saw it,
00:32:41.000 --> 00:32:45.000
but hopefully by now,
well, it should be less
00:32:45.000 --> 00:32:46.000
mysterious.
00:33:07.000 --> 00:33:14.000
OK, so we've learned about the
gradient vector which is del f
00:33:14.000 --> 00:33:21.000
is a vector whose components are
just the partial derivatives.
00:33:21.000 --> 00:33:26.000
So, if I have a function of
just two variables,
00:33:26.000 --> 00:33:29.000
then it's just this.
And,
00:33:29.000 --> 00:33:37.000
so one observation that we've
made is that if you look at a
00:33:37.000 --> 00:33:44.000
contour plot of your function,
so maybe your function is zero,
00:33:44.000 --> 00:33:47.000
one, and two,
then the gradient vector is
00:33:47.000 --> 00:33:49.000
always perpendicular to the
contour plot,
00:33:49.000 --> 00:33:54.000
and always points towards
higher ground.
00:33:54.000 --> 00:34:02.000
OK, so the reason for that was
that if you take any direction,
00:34:02.000 --> 00:34:04.000
you can measure the directional
derivative,
00:34:04.000 --> 00:34:12.000
which means the rate of change
of f in that direction.
00:34:12.000 --> 00:34:20.000
So, given a unit vector, u,
which represents some
00:34:20.000 --> 00:34:24.000
direction,
so for example let's say I
00:34:24.000 --> 00:34:29.000
decide that I want to go in this
direction,
00:34:29.000 --> 00:34:32.000
and I ask myself,
how quickly will f change if I
00:34:32.000 --> 00:34:36.000
start from here and I start
moving towards that direction?
00:34:36.000 --> 00:34:38.000
Well, the answer seems to be,
it will start to increase a
00:34:38.000 --> 00:34:41.000
bit, and maybe at some point
later on something else will
00:34:41.000 --> 00:34:45.000
happen.
But at first, it will increase.
00:34:45.000 --> 00:34:48.000
So,
the directional derivative is
00:34:48.000 --> 00:34:53.000
what we've called f by ds in the
direction of this unit vector,
00:34:53.000 --> 00:34:56.000
and basically the only thing we
know to be able to compute it,
00:34:56.000 --> 00:35:00.000
the only thing we need is that
it's the dot product between the
00:35:00.000 --> 00:35:02.000
gradient and this vector u hat.
In particular,
00:35:02.000 --> 00:35:05.000
the directional derivatives in
the direction of I hat or j hat
00:35:05.000 --> 00:35:07.000
are just the usual partial
derivatives.
00:35:07.000 --> 00:35:12.000
That's what you would expect.
OK, and so now you see in
00:35:12.000 --> 00:35:15.000
particular if you try to go in a
direction that's perpendicular
00:35:15.000 --> 00:35:18.000
to the gradient,
then the directional derivative
00:35:18.000 --> 00:35:21.000
will be zero because you are
moving on the level curve.
00:35:21.000 --> 00:35:27.000
So, the value doesn't change,
OK?
00:35:27.000 --> 00:35:45.000
Questions about that?
Yes?
00:35:45.000 --> 00:35:49.000
Yeah, so let's see,
so indeed to look at more
00:35:49.000 --> 00:35:52.000
recent things,
if you are taking the flux
00:35:52.000 --> 00:35:55.000
through something given by an
equation,
00:35:55.000 --> 00:35:59.000
so, if you have a surface given
by an equation,
00:35:59.000 --> 00:36:05.000
say, f equals one.
So, say that you have a surface
00:36:05.000 --> 00:36:08.000
here or a curve given by an
equation,
00:36:08.000 --> 00:36:14.000
f equals constant,
then the normal vector to the
00:36:14.000 --> 00:36:19.000
surface is given by taking the
gradient of f.
00:36:19.000 --> 00:36:22.000
And that is,
in general, not a unit normal
00:36:22.000 --> 00:36:24.000
vector.
Now, if you wanted the unit
00:36:24.000 --> 00:36:28.000
normal vector to compute flux,
then you would just scale this
00:36:28.000 --> 00:36:30.000
guy down to unit length,
OK?
00:36:30.000 --> 00:36:33.000
So, if you wanted a unit
normal, that would be the
00:36:33.000 --> 00:36:37.000
gradient divided by its length.
However, for flux,
00:36:37.000 --> 00:36:40.000
that's still of limited
usefulness because you would
00:36:40.000 --> 00:36:42.000
still need to know about ds.
But, remember,
00:36:42.000 --> 00:36:46.000
we've seen a formula for flux
in terms of a non-unit normal
00:36:46.000 --> 00:36:52.000
vector, and n over n dot kdxdy.
So, indeed, this is how you
00:36:52.000 --> 00:36:58.000
could actually handle
calculations of flux through
00:36:58.000 --> 00:37:09.000
pretty much anything.
Any other questions about that?
00:37:09.000 --> 00:37:19.000
OK, so let me continue with a
couple more things we need to,
00:37:19.000 --> 00:37:25.000
so, we've seen how to do
min/max problems,
00:37:25.000 --> 00:37:33.000
in particular,
by looking at critical points.
00:37:33.000 --> 00:37:35.000
So, critical points,
remember, are the points where
00:37:35.000 --> 00:37:37.000
all the partial derivatives are
zero.
00:37:37.000 --> 00:37:40.000
So, if you prefer,
that's where the gradient
00:37:40.000 --> 00:37:45.000
vector is zero.
And, we know how to decide
00:37:45.000 --> 00:37:52.000
using the second derivative test
whether a critical point is
00:37:52.000 --> 00:37:57.000
going to be a local min,
a local max,
00:37:57.000 --> 00:38:02.000
or a saddle point.
Actually, we can't always quite
00:38:02.000 --> 00:38:05.000
decide because,
remember, we look at the second
00:38:05.000 --> 00:38:08.000
partials, and we compute this
quantity ac minus b squared.
00:38:08.000 --> 00:38:10.000
And, if it happens to be zero,
then actually we can't
00:38:10.000 --> 00:38:13.000
conclude.
But, most of the time we can
00:38:13.000 --> 00:38:16.000
conclude.
However, that's not all we need
00:38:16.000 --> 00:38:20.000
to look for an absolute global
maximum or minimum.
00:38:20.000 --> 00:38:23.000
For that, we also need to check
the boundary points,
00:38:23.000 --> 00:38:27.000
or look at the behavior of a
function, at infinity.
00:38:27.000 --> 00:38:38.000
So, we also need to check the
values of f at the boundary of
00:38:38.000 --> 00:38:46.000
its domain of definition or at
infinity.
00:38:46.000 --> 00:38:48.000
Just to give you an example
from single variable calculus,
00:38:48.000 --> 00:38:51.000
if you are trying to find the
minimum and the maximum of f of
00:38:51.000 --> 00:38:55.000
x equals x squared,
well, you'll find quickly that
00:38:55.000 --> 00:38:57.000
the minimum is at zero where x
squared is zero.
00:38:57.000 --> 00:39:00.000
If you are looking for the
maximum, you better not just
00:39:00.000 --> 00:39:02.000
look at the derivative because
you won't find it that way.
00:39:02.000 --> 00:39:05.000
However, if you think for a
second, you'll see that if x
00:39:05.000 --> 00:39:08.000
becomes very large,
then the function increases to
00:39:08.000 --> 00:39:10.000
infinity.
And, similarly,
00:39:10.000 --> 00:39:14.000
if you try to find the minimum
and the maximum of x squared
00:39:14.000 --> 00:39:17.000
when x varies only between one
and two,
00:39:17.000 --> 00:39:19.000
well, you won't find the
critical point,
00:39:19.000 --> 00:39:21.000
but you'll still find that the
smallest value of x squared is
00:39:21.000 --> 00:39:24.000
when x is at one,
and the largest is at x equals
00:39:24.000 --> 00:39:26.000
two.
And, all this business about
00:39:26.000 --> 00:39:29.000
boundaries and infinity is
exactly the same stuff,
00:39:29.000 --> 00:39:31.000
but with more than one
variable.
00:39:31.000 --> 00:39:37.000
It's just the story that maybe
the minimum and the maximum are
00:39:37.000 --> 00:39:42.000
not quite visible,
but they are at the edges of a
00:39:42.000 --> 00:39:48.000
domain we are looking at.
Well, in the last three
00:39:48.000 --> 00:39:55.000
minutes, I will just write down
a couple more things we've seen
00:39:55.000 --> 00:40:00.000
there.
So, how to do max/min problems
00:40:00.000 --> 00:40:08.000
with non-independent variables
-- So, if your variables are
00:40:08.000 --> 00:40:15.000
related by some condition,
g equals some constant.
00:40:15.000 --> 00:40:25.000
So, then we've seen the method
of Lagrange multipliers.
00:40:25.000 --> 00:40:31.000
OK, and what this method says
is that we should solve the
00:40:31.000 --> 00:40:36.000
equation gradient f equals some
unknown scalar lambda times the
00:40:36.000 --> 00:40:39.000
gradient, g.
So, that means each partial,
00:40:39.000 --> 00:40:43.000
f sub x equals lambda g sub x
and so on,
00:40:43.000 --> 00:40:48.000
and of course we have to keep
in mind the constraint equation
00:40:48.000 --> 00:40:53.000
so that we have the same number
of equations as the number of
00:40:53.000 --> 00:40:57.000
unknowns because you have a new
unknown here.
00:40:57.000 --> 00:41:04.000
And, the thing to remember is
that you have to be careful that
00:41:04.000 --> 00:41:13.000
the second derivative test does
not apply in this situation.
00:41:13.000 --> 00:41:16.000
I mean, this is only in the
case of independent variables.
00:41:16.000 --> 00:41:18.000
So, if you want to know if
something is a maximum or a
00:41:18.000 --> 00:41:20.000
minimum,
you just have to use common
00:41:20.000 --> 00:41:24.000
sense or compare the values of a
function at the various points
00:41:24.000 --> 00:41:29.000
you found.
Yes?
00:41:29.000 --> 00:41:34.000
Will we actually have to
calculate?
00:41:34.000 --> 00:41:38.000
Well, that depends on what the
problem asks you.
00:41:38.000 --> 00:41:40.000
It might ask you to just set up
the equations,
00:41:40.000 --> 00:41:41.000
or it might ask you to solve
them.
00:41:41.000 --> 00:41:44.000
So, in general,
solving might be difficult,
00:41:44.000 --> 00:41:47.000
but if it asks you to do it,
then it means it shouldn't be
00:41:47.000 --> 00:41:50.000
too hard.
I haven't written the final
00:41:50.000 --> 00:41:54.000
yet, so I don't know what it
will be, but it might be an easy
00:41:54.000 --> 00:42:00.000
one.
And, the last thing we've seen
00:42:00.000 --> 00:42:06.000
is constrained partial
derivatives.
00:42:06.000 --> 00:42:12.000
So, for example,
if you have a relation between
00:42:12.000 --> 00:42:15.000
x, y, and z,
which are constrained to be a
00:42:15.000 --> 00:42:20.000
constant,
then the notion of partial f
00:42:20.000 --> 00:42:24.000
partial x takes several
meanings.
00:42:24.000 --> 00:42:32.000
So, just to remind you very
quickly, there's the formal
00:42:32.000 --> 00:42:38.000
partial, partial f,
partial x, which means x
00:42:38.000 --> 00:42:43.000
varies.
Y and z are held constant.
00:42:43.000 --> 00:42:48.000
And, we forget the constraint.
This is not compatible with a
00:42:48.000 --> 00:42:51.000
constraint, but we don't care.
So, that's the guy that we
00:42:51.000 --> 00:42:54.000
compute just from the formula
for f ignoring the constraints.
00:42:54.000 --> 00:43:01.000
And then, we have the partial
f, partial x with y held
00:43:01.000 --> 00:43:06.000
constant, which means y held
constant.
00:43:06.000 --> 00:43:15.000
X varies, and now we treat z as
a dependent variable.
00:43:15.000 --> 00:43:20.000
It varies with x and y
according to whatever is needed
00:43:20.000 --> 00:43:24.000
so that this constraint keeps
holding.
00:43:24.000 --> 00:43:29.000
And, similarly,
there's partial f partial x
00:43:29.000 --> 00:43:33.000
with z held constant,
which means that,
00:43:33.000 --> 00:43:38.000
now, y is the dependent
variable.
00:43:38.000 --> 00:43:39.000
And, the way in which we
compute these,
00:43:39.000 --> 00:43:42.000
we've seen two methods which
I'm not going to tell you now
00:43:42.000 --> 00:43:45.000
because otherwise we'll be even
more over time.
00:43:45.000 --> 00:43:48.000
But, we've seen two methods for
computing these based on either
00:43:48.000 --> 00:43:50.000
the chain rule or on
differentials,
00:43:50.000 --> 00:43:52.000
solving and substituting into
differentials.