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OK, so anyway,
let's get started.

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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

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class on Tuesday just because we
have to start somewhere.

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So, the first things that we
learned about in this class were

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vectors, and how to do
dot-product of vectors.

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So, remember the formula that A
dot B is the sum of ai times bi.

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And, geometrically,
it's length A times length B

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00:01:05,000 --> 00:01:08,000
times the cosine of the angle
between them.

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00:01:08,000 --> 00:01:11,000
And, in particular,
we can use this to detect when

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00:01:11,000 --> 00:01:14,000
two vectors are perpendicular.
That's when their dot product

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is zero.
And, we can use that to measure

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angles between vectors by
solving for cosine in this.

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00:01:21,000 --> 00:01:25,000
Hopefully, at this point,
this looks a lot easier than it

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used to a few months ago.
So, hopefully at this point,

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everyone has this kind of
formula memorized and has some

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reasonable understanding of
that.

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00:01:35,000 --> 00:01:41,000
But, if you have any questions,
now is the time.

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00:01:41,000 --> 00:01:45,000
No?
Good.

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00:01:45,000 --> 00:01:55,000
Next we learned how to also do
cross product of vectors in

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space -- -- and remember,
we saw how to use that to find

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00:02:06,000 --> 00:02:10,000
area of, say,
a triangle or a parallelogram

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00:02:10,000 --> 00:02:14,000
in space because the length of
the cross product is equal to

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00:02:14,000 --> 00:02:17,000
the area of a parallelogram
formed by the vectors a and b.

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00:02:17,000 --> 00:02:25,000
And, we can also use that to
find a vector perpendicular to

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two given vectors,
A and B.

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And so, in particular,
that comes in handy when we are

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00:02:33,000 --> 00:02:42,000
looking for the equation of a
plane because we've seen -- So,

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00:02:42,000 --> 00:02:49,000
the next topic would be
equations of planes.

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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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00:02:55,000 --> 00:02:59,000
the form ax by cz = d,
well, <a,

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00:02:59,000 --> 00:03:03,000
b, c> in there is actually
the normal vector to the plane,

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00:03:03,000 --> 00:03:07,000
or some normal vector to the
plane.

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00:03:07,000 --> 00:03:11,000
So, typically,
we use cross product to find

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plane equations.
OK, is that still reasonably

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familiar to everyone?
Yes, very good.

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OK, we've also seen how to look
at equations of lines,

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00:03:26,000 --> 00:03:31,000
and those were of a slightly
different nature because we've

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00:03:31,000 --> 00:03:35,000
been doing them as parametric
equations.

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00:03:35,000 --> 00:03:42,000
So, typically we had equations
of a form, maybe x equals some

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constant times t,
y equals constant plus constant

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00:03:47,000 --> 00:03:53,000
times t.
z equals constant plus constant

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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

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00:04:06,000 --> 00:04:11,000
correspond to a vector parallel
to the line.

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00:04:11,000 --> 00:04:19,000
That's the velocity of the
moving point on the line.

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00:04:19,000 --> 00:04:23,000
And, well,
we've learned in particular how

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to find where a line intersects
a plane by plugging in the

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00:04:29,000 --> 00:04:34,000
parametric equation into the
equation of a plane.

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We've learned more general
things about parametric

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equations of curves.
So, there are these infamous

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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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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

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wheel that's itself moving and
something else,

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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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00:05:14,000 --> 00:05:18,000
turned,
and then get to the position of

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00:05:18,000 --> 00:05:23,000
a moving point by adding
together simpler vectors.

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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,

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00:05:30,000 --> 00:05:36,000
and then decompose the motion
into a sum of simpler effect.

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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

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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,

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yes?
Sorry? What about it?

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00:06:24,000 --> 00:06:25,000
Parametric descriptions of a
plane,

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so we haven't really done that
because you would need two

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parameters to parameterize a
plane just because it's a two

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dimensional object.
So, we have mostly focused on

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00:06:35,000 --> 00:06:40,000
the use of parametric equations
just for one dimensional

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00:06:40,000 --> 00:06:42,000
objects, lines,
and curves.

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So,
you won't need to know about

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00:06:45,000 --> 00:06:47,000
parametric descriptions of
planes on a final,

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00:06:47,000 --> 00:06:51,000
but if you really wanted to,
you would think of defining a

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point on a plane as starting
from some given point.

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00:06:55,000 --> 00:06:57,000
Then you have two vectors given
on the plane.

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And then, you would add a
multiple of each of these

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00:07:00,000 --> 00:07:04,000
vectors to your starting point.
But see, the difficulty is to

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00:07:04,000 --> 00:07:08,000
convert from that to the usual
equation of a plane,

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00:07:08,000 --> 00:07:11,000
you would still have to go back
to this cross product method,

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00:07:11,000 --> 00:07:15,000
and so on.
So, it is possible to represent

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a plane, or, in general,
a surface in parametric form.

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But, very often,
that's not so useful.

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Yes?
How do you parametrize an

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ellipse in space?
Well, that depends on how it's

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given to you.
But, OK, let's just do an

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example.
Say that I give you an ellipse

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00:07:38,000 --> 00:07:42,000
in space as maybe the more,
well, one exciting way to

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parameterize an ellipse in space
is maybe the intersection of a

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cylinder with a slanted plane.
That's the kind of situations

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where you might end up with an
ellipse.

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00:07:52,000 --> 00:07:58,000
OK, so if I tell you that maybe
I'm intersecting a cylinder with

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equation x squared plus y
squared equals a squared with a

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00:08:03,000 --> 00:08:09,000
slanted plane to get,
I messed up my picture,

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00:08:09,000 --> 00:08:13,000
to get this ellipse of
intersection,

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so, of course you'd need the
equation of a plane.

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And, let's say that this plane
is maybe given to you.

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Or, you can switch it to form
where you can get z as a

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function of x and y.
So, maybe it would be z equals,

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I've already used a;
I need to use a new letter.

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00:08:33,000 --> 00:08:41,000
Let's say c1x c2y plus d,
whatever, something like that.

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00:08:41,000 --> 00:08:45,000
So, what I would do is first I
would look at what my ellipse

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does in the directions in which
I understand it the best.

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00:08:49,000 --> 00:08:53,000
And, those directions would be
probably the xy plane.

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So, I would look at the xy
coordinates.

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Well, if I look at it from
above xy, my ellipse looks like

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00:09:02,000 --> 00:09:06,000
just a circle of radius a.
So, if I'm only concerned with

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00:09:06,000 --> 00:09:10,000
x and y, presumably I can just
do it the usual way for a

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00:09:10,000 --> 00:09:13,000
circle.
x equals a cosine t.

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00:09:13,000 --> 00:09:20,000
y equals a sine t, OK?
And then, z would end up being

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just, well, whatever the value
of z is to be on the slanted

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plane above a given xy position.
So, in fact,

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it would end up being ac1
cosine t plus ac2 sine t plus d,

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I guess.
OK, that's not a particularly

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elegant parameterization,
but that's the kind of thing

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you might end up with.
Now, in general,

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00:09:47,000 --> 00:09:50,000
when you have a curve in space,
it would rarely be the case

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00:09:50,000 --> 00:09:53,000
that you have to get a
parameterization from scratch

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unless you are already being
told information about how it

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looks in one of the coordinate
planes,

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00:09:58,000 --> 00:10:03,000
this kind of method.
Or, at least you'd have a lot

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00:10:03,000 --> 00:10:07,000
of information that would
quickly reduce to a plane

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00:10:07,000 --> 00:10:11,000
problem somehow.
Of course, I could also just

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00:10:11,000 --> 00:10:16,000
give you some formulas and let
you figure out what's going on.

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00:10:16,000 --> 00:10:21,000
But, in general,
we've done more stuff with

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plane curves.
With plane curves,

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certainly there's interesting
things with all sorts of

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00:10:29,000 --> 00:10:32,000
mechanical gadgets that we can
study.

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00:10:32,000 --> 00:10:39,000
OK, any other questions on that?
No?

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00:10:39,000 --> 00:10:45,000
OK, so let me move on a bit and
point out that with parametric

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00:10:45,000 --> 00:10:51,000
equations, we've looked also at
things like velocity and

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00:10:51,000 --> 00:10:55,000
acceleration.
So, the velocity vector is the

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00:10:55,000 --> 00:10:59,000
derivative of a position vector
with respect to time.

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00:10:59,000 --> 00:11:04,000
And, it's not to be confused
with speed, which is the

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00:11:04,000 --> 00:11:08,000
magnitude of v.
So, the velocity vector is

148
00:11:08,000 --> 00:11:12,000
going to be always tangent to
the curve.

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00:11:12,000 --> 00:11:14,000
And, its length will be the
speed.

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00:11:14,000 --> 00:11:15,000
That's the geometric
interpretation.

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00:11:32,000 --> 00:11:37,000
So, just to provoke you,
I'm going to write,

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00:11:37,000 --> 00:11:43,000
again, that formula that was
that v equals T hat ds dt.

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00:11:43,000 --> 00:11:46,000
What do I mean by that?
If I have a curve,

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00:11:46,000 --> 00:11:51,000
and I'm moving on the curve,
well, I have the unit tangent

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00:11:51,000 --> 00:11:56,000
vector which I think at the time
I used to draw in blue.

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00:11:56,000 --> 00:11:59,000
But, blue has been abolished
since then.

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00:11:59,000 --> 00:12:04,000
So, I'm going to draw it in red.
OK, so that's a unit vector

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00:12:04,000 --> 00:12:09,000
that goes along the curve,
and then the actual velocity is

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00:12:09,000 --> 00:12:11,000
going to be proportional to
that.

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00:12:11,000 --> 00:12:15,000
And, what's the length?
Well, it's the speed.

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00:12:15,000 --> 00:12:19,000
And, the speed is how much arc
length on the curve I go per

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00:12:19,000 --> 00:12:22,000
unit time, which is why I'm
writing ds dt.

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00:12:22,000 --> 00:12:30,000
That's another guy.
That's another of these guys

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00:12:30,000 --> 00:12:34,000
for the speed,
OK?

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00:12:34,000 --> 00:12:41,000
And, we've also learned about
acceleration,

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00:12:41,000 --> 00:12:47,000
which is the derivative of
velocity.

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00:12:47,000 --> 00:12:50,000
So, it's the second derivative
of a position vector.

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00:12:50,000 --> 00:12:54,000
And, as an example of the kinds
of manipulations we can do,

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00:12:54,000 --> 00:12:56,000
in class we've seen Kepler's
second law,

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00:12:56,000 --> 00:13:03,000
which explains how if the
acceleration is parallel to the

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00:13:03,000 --> 00:13:08,000
position vector,
then r cross v is going to be

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00:13:08,000 --> 00:13:10,000
constant,
which means that the motion

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00:13:10,000 --> 00:13:13,000
will be in an plane,
and you will sweep area at a

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00:13:13,000 --> 00:13:16,000
constant rate.
So now, that is not in itself a

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00:13:16,000 --> 00:13:19,000
topic for the exam,
but the kinds of methods of

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00:13:19,000 --> 00:13:22,000
differentiating vector
quantities,

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00:13:22,000 --> 00:13:25,000
applying the product rule to
take the derivative of a dot or

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00:13:25,000 --> 00:13:28,000
cross product and so on are
definitely fair game.

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00:13:28,000 --> 00:13:30,000
I mean, we've seen those on the
first exam.

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00:13:30,000 --> 00:13:35,000
They were there,
and most likely they will be on

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00:13:35,000 --> 00:13:39,000
the final.
OK, so I mean that's the extent

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00:13:39,000 --> 00:13:44,000
to which Kepler's law comes up,
only just knowing the general

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00:13:44,000 --> 00:13:47,000
type of manipulations and
proving things with vector

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00:13:47,000 --> 00:13:52,000
quantities,
but not again the actual

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00:13:52,000 --> 00:13:58,000
Kepler's law itself.
I skipped something.

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00:13:58,000 --> 00:14:08,000
I skipped matrices,
determinants,

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00:14:08,000 --> 00:14:18,000
and linear systems.
OK, so we've seen how to

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00:14:18,000 --> 00:14:24,000
multiply matrices,
and how to write linear systems

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00:14:24,000 --> 00:14:28,000
in matrix form.
So, remember,

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00:14:28,000 --> 00:14:35,000
if you have a 3x3 linear system
in the usual sense,

191
00:14:35,000 --> 00:14:42,000
so,
you can write this in a matrix

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00:14:42,000 --> 00:14:52,000
form where you have a 3x3 matrix
and you have an unknown column

193
00:14:52,000 --> 00:14:57,000
vector.
And, their matrix product

194
00:14:57,000 --> 00:15:01,000
should be some given column
vector.

195
00:15:01,000 --> 00:15:04,000
OK, so if you don't remember
how to multiply matrices,

196
00:15:04,000 --> 00:15:07,000
please look at the notes on
that again.

197
00:15:07,000 --> 00:15:12,000
And, also you should remember
how to invert a matrix.

198
00:15:12,000 --> 00:15:22,000
So, how did we invert matrices?
Let me just remind you very

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00:15:22,000 --> 00:15:30,000
quickly.
So, I should say 2x2 or 3x3

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00:15:30,000 --> 00:15:33,000
matrices.
Well, you need to have a square

201
00:15:33,000 --> 00:15:35,000
matrix to be able to find an
inverse.

202
00:15:35,000 --> 00:15:37,000
The method doesn't work,
doesn't make sense.

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00:15:37,000 --> 00:15:40,000
Otherwise, then the concept of
inverse doesn't work.

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00:15:40,000 --> 00:15:43,000
And, if it's larger than 3x3,
then we haven't seen that.

205
00:15:43,000 --> 00:15:50,000
So, let's say that I have a 3x3
matrix.

206
00:15:50,000 --> 00:16:00,000
What I will do is I will start
by forming the matrix of minors.

207
00:16:00,000 --> 00:16:09,000
So, remember that minors,
so, each entry is a 2x2

208
00:16:09,000 --> 00:16:20,000
determinant in the case of a 3x3
matrix formed by deleting one

209
00:16:20,000 --> 00:16:26,000
row and one column.
OK, so for example,

210
00:16:26,000 --> 00:16:30,000
to get the first minor,
especially in the upper left

211
00:16:30,000 --> 00:16:34,000
corner, I would delete the first
row, the first column.

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00:16:34,000 --> 00:16:36,000
And, I would be left with this
2x2 determinant.

213
00:16:36,000 --> 00:16:38,000
I take this times that minus
this times that.

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00:16:38,000 --> 00:16:41,000
I get a number that gives my
first minor.

215
00:16:41,000 --> 00:16:49,000
And then, same with the others.
Then, I flip signs according to

216
00:16:49,000 --> 00:16:56,000
this checkerboard pattern,
and that gives me the matrix of

217
00:16:56,000 --> 00:17:00,000
cofactors.
OK, so all it means is I'm just

218
00:17:00,000 --> 00:17:06,000
changing the signs of these four
entries and leaving the others

219
00:17:06,000 --> 00:17:10,000
alone.
And then, I take the transpose

220
00:17:10,000 --> 00:17:13,000
of that.
So, that means I read it

221
00:17:13,000 --> 00:17:16,000
horizontally and write it down
vertically.

222
00:17:16,000 --> 00:17:19,000
I swept the rows and the
columns.

223
00:17:19,000 --> 00:17:23,000
And then, I divide by the
inverse.

224
00:17:23,000 --> 00:17:28,000
Well, I divide by the
determinant of the initial

225
00:17:28,000 --> 00:17:30,000
matrix.
OK, so, of course,

226
00:17:30,000 --> 00:17:32,000
this is kind of very
theoretical, and I write it like

227
00:17:32,000 --> 00:17:34,000
this.
Probably it makes more sense to

228
00:17:34,000 --> 00:17:37,000
do it on an example.
I will let you work out

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00:17:37,000 --> 00:17:42,000
examples, or bug your recitation
instructors so that they do one

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00:17:42,000 --> 00:17:44,000
on Monday if you want to see
that.

231
00:17:44,000 --> 00:17:47,000
It's a fairly straightforward
method.

232
00:17:47,000 --> 00:17:50,000
You just have to remember the
steps.

233
00:17:50,000 --> 00:17:52,000
But, of course,
there's one condition,

234
00:17:52,000 --> 00:17:57,000
which is that the determinant
of a matrix has to be nonzero.

235
00:17:57,000 --> 00:17:59,000
So, in fact,
we've seen that,

236
00:17:59,000 --> 00:18:03,000
oh, there is still one board
left.

237
00:18:03,000 --> 00:18:12,000
We've seen that a matrix is
invertible -- -- exactly when

238
00:18:12,000 --> 00:18:19,000
its determinant is not zero.
And, if that's the case,

239
00:18:19,000 --> 00:18:24,000
then we can solve the linear
system, AX equals B by just

240
00:18:24,000 --> 00:18:30,000
setting X equals A inverse B.
That's going to be the only

241
00:18:30,000 --> 00:18:38,000
solution to our linear system.
Otherwise, well,

242
00:18:38,000 --> 00:18:52,000
AX equals B has either no
solution, or infinitely many

243
00:18:52,000 --> 00:19:01,000
solutions.
Yes?

244
00:19:01,000 --> 00:19:04,000
The determinant of a matrix
real quick?

245
00:19:04,000 --> 00:19:08,000
Well, I can do it that quickly
unless I start waving my hands

246
00:19:08,000 --> 00:19:12,000
very quickly,
but remember we've seen that

247
00:19:12,000 --> 00:19:15,000
you have a matrix,
a 3x3 matrix.

248
00:19:15,000 --> 00:19:18,000
Its determinant will be
obtained by doing an expansion

249
00:19:18,000 --> 00:19:20,000
with respect to,
well, your favorite.

250
00:19:20,000 --> 00:19:22,000
But usually,
we are doing it with respect to

251
00:19:22,000 --> 00:19:26,000
the first row.
So, we take this entry and

252
00:19:26,000 --> 00:19:31,000
multiply it by that determinant.
Then, we take that entry,

253
00:19:31,000 --> 00:19:35,000
multiply it by that determinant
but put a minus sign.

254
00:19:35,000 --> 00:19:38,000
And then, we take that entry
and multiply it by this

255
00:19:38,000 --> 00:19:41,000
determinant here,
and we put a plus sign for

256
00:19:41,000 --> 00:19:44,000
that.
OK, so maybe I should write it

257
00:19:44,000 --> 00:19:46,000
down.
That's actually the same

258
00:19:46,000 --> 00:19:48,000
formula that we are using for
cross products.

259
00:19:48,000 --> 00:19:50,000
Right, when we do cross
products, we are doing an

260
00:19:50,000 --> 00:19:53,000
expansion with respect to the
first row.

261
00:19:53,000 --> 00:19:57,000
That's a special case.
OK, I mean, do you still want

262
00:19:57,000 --> 00:19:59,000
to see it in more details,
or is that OK?

263
00:19:59,000 --> 00:20:12,000
Yes?
That's correct.

264
00:20:12,000 --> 00:20:16,000
So, if you do an expansion with
respect to any row or column,

265
00:20:16,000 --> 00:20:19,000
then you would use the same
signs that are in this

266
00:20:19,000 --> 00:20:22,000
checkerboard pattern there.
So, if you did an expansion,

267
00:20:22,000 --> 00:20:25,000
actually, so indeed,
maybe I should say,

268
00:20:25,000 --> 00:20:28,000
the more general way to
determine it is you take your

269
00:20:28,000 --> 00:20:31,000
favorite row or column,
and you just multiply the

270
00:20:31,000 --> 00:20:34,000
corresponding entries by the
corresponding cofactors.

271
00:20:34,000 --> 00:20:37,000
So, the signs are plus or minus
depending on what's in that

272
00:20:37,000 --> 00:20:38,000
diagram there.
Now, in practice,

273
00:20:38,000 --> 00:20:41,000
in this class,
again, all we need is to do it

274
00:20:41,000 --> 00:20:46,000
with respect to the first row.
So, don't worry about it too

275
00:20:46,000 --> 00:20:48,000
much.
OK, so, again,

276
00:20:48,000 --> 00:20:51,000
the way that we've officially
seen it in this class is just if

277
00:20:51,000 --> 00:20:59,000
you have a1,
a2, a3, b1, b2, b3, c1, c2, c3,

278
00:20:59,000 --> 00:21:06,000
so if the determinant is a1
times b2 b3, c2 c3,

279
00:21:06,000 --> 00:21:16,000
minus a2 b1 b3 c1 c3 plus a3 b1
b2 c1 c2.

280
00:21:16,000 --> 00:21:20,000
And, this minus is here
basically because of the minus

281
00:21:20,000 --> 00:21:27,000
in the diagram up there.
But, that's all we need to know.

282
00:21:27,000 --> 00:21:32,000
Yes?
How do you tell the difference

283
00:21:32,000 --> 00:21:34,000
between infinitely many
solutions or no solutions?

284
00:21:34,000 --> 00:21:37,000
That's a very good question.
So, in full generality,

285
00:21:37,000 --> 00:21:40,000
the answer is we haven't quite
seen a systematic method.

286
00:21:40,000 --> 00:21:43,000
So, you just have to try
solving and see if you can find

287
00:21:43,000 --> 00:21:46,000
a solution or not.
So, let me actually explain

288
00:21:46,000 --> 00:21:51,000
that more carefully.
So, what happens to these two

289
00:21:51,000 --> 00:21:56,000
situations when a is invertible
or not?

290
00:21:56,000 --> 00:21:57,000
So, remember,
in the linear system,

291
00:21:57,000 --> 00:22:01,000
you can think of a linear
system as asking you to find the

292
00:22:01,000 --> 00:22:05,000
intersection between three
planes because each equation is

293
00:22:05,000 --> 00:22:12,000
the equation of a plane.
So, Ax = B for a 3x3 system

294
00:22:12,000 --> 00:22:24,000
means that x should be in the
intersection of three planes.

295
00:22:24,000 --> 00:22:28,000
And then, we have two cases.
So, the case where the system

296
00:22:28,000 --> 00:22:33,000
is invertible corresponds to the
general situation where your

297
00:22:33,000 --> 00:22:37,000
three planes somehow all just
intersect in one point.

298
00:22:37,000 --> 00:22:41,000
And then, the situation where
the determinant,

299
00:22:41,000 --> 00:22:45,000
that's when the determinant is
not zero, you get just one

300
00:22:45,000 --> 00:22:48,000
point.
However, sometimes it will

301
00:22:48,000 --> 00:22:54,000
happen that all the planes are
parallel to the same direction.

302
00:22:54,000 --> 00:23:04,000
So, determinant a equals zero
means the three planes are

303
00:23:04,000 --> 00:23:11,000
parallel to a same vector.
And, in fact,

304
00:23:11,000 --> 00:23:14,000
you can find that vector
explicitly because that vector

305
00:23:14,000 --> 00:23:17,000
has to be perpendicular to all
the normals.

306
00:23:17,000 --> 00:23:22,000
So, at some point we saw other
subtle things about how to find

307
00:23:22,000 --> 00:23:26,000
the direction of this line
that's parallel to all the

308
00:23:26,000 --> 00:23:30,000
planes.
So, now, this can happen either

309
00:23:30,000 --> 00:23:34,000
with all three planes containing
the same line.

310
00:23:34,000 --> 00:23:36,000
You know, they can all pass
through the same axis.

311
00:23:36,000 --> 00:23:39,000
Or it could be that they have
somehow shifted with respect to

312
00:23:39,000 --> 00:23:44,000
each other.
And so, it might look like this.

313
00:23:44,000 --> 00:23:46,000
Then, the last one is actually
in front of that.

314
00:23:46,000 --> 00:23:52,000
So, see, the lines of
intersections between two of the

315
00:23:52,000 --> 00:23:55,000
planes,
so, here they all pass through

316
00:23:55,000 --> 00:23:57,000
the same line,
and here, instead,

317
00:23:57,000 --> 00:24:00,000
they intersect in one line
here,

318
00:24:00,000 --> 00:24:03,000
one line here,
and one line there.

319
00:24:03,000 --> 00:24:06,000
And, there's no triple
intersection.

320
00:24:06,000 --> 00:24:08,000
So, in general,
we haven't really seen how to

321
00:24:08,000 --> 00:24:13,000
decide between these two cases.
There's one important situation

322
00:24:13,000 --> 00:24:20,000
where we have seen we must be in
the first case that when we have

323
00:24:20,000 --> 00:24:26,000
a homogeneous system,
so that means if the right hand

324
00:24:26,000 --> 00:24:31,000
side is zero,
then,

325
00:24:31,000 --> 00:24:41,000
well, x equals zero is always a
solution.

326
00:24:41,000 --> 00:24:43,000
It's called the trivial
solution.

327
00:24:43,000 --> 00:24:50,000
It's the obvious one,
if you want.

328
00:24:50,000 --> 00:24:53,000
So, you know that,
and why is that?

329
00:24:53,000 --> 00:24:57,000
Well, that's because all of
your planes have to pass through

330
00:24:57,000 --> 00:25:00,000
the origin.
So, you must be in this case if

331
00:25:00,000 --> 00:25:04,000
you have a noninvertible system
where the right hand side is

332
00:25:04,000 --> 00:25:05,000
zero.
So, in that case,

333
00:25:05,000 --> 00:25:08,000
if the right hand side is zero,
there's two cases.

334
00:25:08,000 --> 00:25:12,000
Either the matrix is invertible.
Then, the only solution is the

335
00:25:12,000 --> 00:25:14,000
trivial one.
Or, if a matrix is not

336
00:25:14,000 --> 00:25:19,000
invertible, then you have
infinitely many solutions.

337
00:25:19,000 --> 00:25:23,000
If B is not zero,
then we haven't really seen how

338
00:25:23,000 --> 00:25:27,000
to decide.
We've just seen how to decide

339
00:25:27,000 --> 00:25:30,000
between one solution or
zero,infinitely many,

340
00:25:30,000 --> 00:25:33,000
but not how to decide between
these last two cases.

341
00:25:33,000 --> 00:25:42,000
Yes?
I think in principle,

342
00:25:42,000 --> 00:25:44,000
you would be able to,
but that's, well,

343
00:25:44,000 --> 00:25:48,000
I mean, that's a slightly
counterintuitive way of doing

344
00:25:48,000 --> 00:25:50,000
it.
I think it would probably work.

345
00:25:50,000 --> 00:25:55,000
Well, I'll let you figure it
out.

346
00:25:55,000 --> 00:25:59,000
OK, let me move on to the
second unit, maybe,

347
00:25:59,000 --> 00:26:03,000
because we've seen a lot of
stuff, or was there a quick

348
00:26:03,000 --> 00:26:05,000
question before that?
OK.

349
00:26:41,000 --> 00:26:44,000
OK, so what was the second part
of the class about?

350
00:26:44,000 --> 00:26:47,000
Well, hopefully you kind of
vaguely remember that it was

351
00:26:47,000 --> 00:26:50,000
about functions of several
variables and their partial

352
00:26:50,000 --> 00:26:55,000
derivatives.
OK, so the first thing that

353
00:26:55,000 --> 00:27:04,000
we've seen is how to actually
view a function of two variables

354
00:27:04,000 --> 00:27:12,000
in terms of its graph and its
contour plot.

355
00:27:12,000 --> 00:27:15,000
So,
just to remind you very

356
00:27:15,000 --> 00:27:17,000
quickly,
if I have a function of two

357
00:27:17,000 --> 00:27:21,000
variables, x and y,
then the graph will be just the

358
00:27:21,000 --> 00:27:25,000
surface given by the equation z
equals f of xy.

359
00:27:25,000 --> 00:27:28,000
So, for each x and y,
I plot a point at height given

360
00:27:28,000 --> 00:27:30,000
with the value of the a
function.

361
00:27:30,000 --> 00:27:34,000
And then, the contour plot will
be the topographical map for

362
00:27:34,000 --> 00:27:37,000
this graph.
It will tell us,

363
00:27:37,000 --> 00:27:41,000
what are the various levels in
there?

364
00:27:41,000 --> 00:27:46,000
So, what it amounts to is we
slice the graph by horizontal

365
00:27:46,000 --> 00:27:50,000
planes, and we get a bunch of
curves which are the points at

366
00:27:50,000 --> 00:27:56,000
given height on the plot.
And, so we get all of these

367
00:27:56,000 --> 00:28:04,000
curves, and then we look at them
from above, and that gives us

368
00:28:04,000 --> 00:28:09,000
this map with a bunch of curves
on it.

369
00:28:09,000 --> 00:28:13,000
And, each of them has a number
next to it which tells us the

370
00:28:13,000 --> 00:28:16,000
value of a function there.
And, from that map, we can,

371
00:28:16,000 --> 00:28:19,000
of course, tell things about
where we might be able to find

372
00:28:19,000 --> 00:28:22,000
minima or maxima of our
function,

373
00:28:22,000 --> 00:28:30,000
and how it varies with respect
to x or y or actually in any

374
00:28:30,000 --> 00:28:40,000
direction at a given point.
So, now, the next thing that

375
00:28:40,000 --> 00:28:49,000
we've learned about is partial
derivatives.

376
00:28:49,000 --> 00:28:52,000
So, for a function of two
variables, there would be two of

377
00:28:52,000 --> 00:28:54,000
them.
There's f sub x which is

378
00:28:54,000 --> 00:28:58,000
partial f partial x,
and f sub y which is partial f

379
00:28:58,000 --> 00:29:00,000
partial y.
And, in terms of a graph,

380
00:29:00,000 --> 00:29:04,000
they correspond to slicing by a
plane that's parallel to one of

381
00:29:04,000 --> 00:29:07,000
the coordinate planes,
so that we either keep x

382
00:29:07,000 --> 00:29:10,000
constant,
or keep y constant.

383
00:29:10,000 --> 00:29:14,000
And, we look at the slope of a
graph to see the rate of change

384
00:29:14,000 --> 00:29:17,000
of f with respect to one
variable only when we hold the

385
00:29:17,000 --> 00:29:21,000
other one constant.
And so, we've seen in

386
00:29:21,000 --> 00:29:25,000
particular how to use that in
various places,

387
00:29:25,000 --> 00:29:29,000
but, for example,
for linear approximation we've

388
00:29:29,000 --> 00:29:34,000
seen that the change in f is
approximately equal to f sub x

389
00:29:34,000 --> 00:29:40,000
times the change in x plus f sub
y times the change in y.

390
00:29:40,000 --> 00:29:45,000
So, you can think of f sub x
and f sub y as telling you how

391
00:29:45,000 --> 00:29:49,000
sensitive the value of f is to
changes in x and y.

392
00:29:49,000 --> 00:29:59,000
So, this linear approximation
also tells us about the tangent

393
00:29:59,000 --> 00:30:07,000
plane to the graph of f.
In fact, when we turn this into

394
00:30:07,000 --> 00:30:16,000
an equality, that would mean
that we replace f by the tangent

395
00:30:16,000 --> 00:30:19,000
plane.
We've also learned various ways

396
00:30:19,000 --> 00:30:21,000
of, before I go on,
I should say,

397
00:30:21,000 --> 00:30:24,000
of course, we've seen these
also for functions of three

398
00:30:24,000 --> 00:30:28,000
variables, right?
So, we haven't seen how to plot

399
00:30:28,000 --> 00:30:32,000
them, and we don't really worry
about that too much.

400
00:30:32,000 --> 00:30:37,000
But, if you have a function of
three variables,

401
00:30:37,000 --> 00:30:42,000
you can do the same kinds of
manipulations.

402
00:30:42,000 --> 00:30:49,000
So, we've learned about
differentials and chain rules,

403
00:30:49,000 --> 00:30:57,000
which are a way of repackaging
these partial derivatives.

404
00:30:57,000 --> 00:31:00,000
So, the differential is just,
by definition,

405
00:31:00,000 --> 00:31:05,000
this thing called df which is f
sub x times dx plus f sub y

406
00:31:05,000 --> 00:31:09,000
times dy.
And, what we can do with it is

407
00:31:09,000 --> 00:31:14,000
just either plug values for
changes in x and y,

408
00:31:14,000 --> 00:31:17,000
and get approximation formulas,
or we can look at this in a

409
00:31:17,000 --> 00:31:21,000
situation where x and y will
depend on something else,

410
00:31:21,000 --> 00:31:26,000
and we get a chain rule.
So, for example,

411
00:31:26,000 --> 00:31:32,000
if f is a function of t time,
for example, and so is y,

412
00:31:32,000 --> 00:31:36,000
then we can find the rate of
change of f with respect to t

413
00:31:36,000 --> 00:31:43,000
just by dividing this by dt.
So, we get df dt equals f sub x

414
00:31:43,000 --> 00:31:48,000
dx dt plus f sub y dy dt.
We can also get other chain

415
00:31:48,000 --> 00:31:51,000
rules,
say, if x and y depend on more

416
00:31:51,000 --> 00:31:54,000
than one variable,
if you have a change of

417
00:31:54,000 --> 00:31:55,000
variables,
for example,

418
00:31:55,000 --> 00:31:58,000
x and y are functions of two
other guys that you call u and

419
00:31:58,000 --> 00:32:01,000
v,
then you can express dx and dy

420
00:32:01,000 --> 00:32:05,000
in terms of du and dv,
and plugging into df you will

421
00:32:05,000 --> 00:32:08,000
get the manner in which f
depends on u and v.

422
00:32:08,000 --> 00:32:11,000
So, that will give you formulas
for partial f partial u,

423
00:32:11,000 --> 00:32:14,000
and partial f partial v.
They look just like these guys

424
00:32:14,000 --> 00:32:19,000
except there's a lot of curly
d's instead of straight ones,

425
00:32:19,000 --> 00:32:21,000
and u's and v's in the
denominators.

426
00:32:21,000 --> 00:32:26,000
OK, so that lets us understand
rates of change.

427
00:32:26,000 --> 00:32:31,000
We've also seen yet another way
to package partial derivatives

428
00:32:31,000 --> 00:32:33,000
into not a differential,
but instead,

429
00:32:33,000 --> 00:32:37,000
a vector.
That's the gradient vector,

430
00:32:37,000 --> 00:32:41,000
and I'm sure it was quite
mysterious when we first saw it,

431
00:32:41,000 --> 00:32:45,000
but hopefully by now,
well, it should be less

432
00:32:45,000 --> 00:32:46,000
mysterious.

433
00:33:07,000 --> 00:33:14,000
OK, so we've learned about the
gradient vector which is del f

434
00:33:14,000 --> 00:33:21,000
is a vector whose components are
just the partial derivatives.

435
00:33:21,000 --> 00:33:26,000
So, if I have a function of
just two variables,

436
00:33:26,000 --> 00:33:29,000
then it's just this.
And,

437
00:33:29,000 --> 00:33:37,000
so one observation that we've
made is that if you look at a

438
00:33:37,000 --> 00:33:44,000
contour plot of your function,
so maybe your function is zero,

439
00:33:44,000 --> 00:33:47,000
one, and two,
then the gradient vector is

440
00:33:47,000 --> 00:33:49,000
always perpendicular to the
contour plot,

441
00:33:49,000 --> 00:33:54,000
and always points towards
higher ground.

442
00:33:54,000 --> 00:34:02,000
OK, so the reason for that was
that if you take any direction,

443
00:34:02,000 --> 00:34:04,000
you can measure the directional
derivative,

444
00:34:04,000 --> 00:34:12,000
which means the rate of change
of f in that direction.

445
00:34:12,000 --> 00:34:20,000
So, given a unit vector, u,
which represents some

446
00:34:20,000 --> 00:34:24,000
direction,
so for example let's say I

447
00:34:24,000 --> 00:34:29,000
decide that I want to go in this
direction,

448
00:34:29,000 --> 00:34:32,000
and I ask myself,
how quickly will f change if I

449
00:34:32,000 --> 00:34:36,000
start from here and I start
moving towards that direction?

450
00:34:36,000 --> 00:34:38,000
Well, the answer seems to be,
it will start to increase a

451
00:34:38,000 --> 00:34:41,000
bit, and maybe at some point
later on something else will

452
00:34:41,000 --> 00:34:45,000
happen.
But at first, it will increase.

453
00:34:45,000 --> 00:34:48,000
So,
the directional derivative is

454
00:34:48,000 --> 00:34:53,000
what we've called f by ds in the
direction of this unit vector,

455
00:34:53,000 --> 00:34:56,000
and basically the only thing we
know to be able to compute it,

456
00:34:56,000 --> 00:35:00,000
the only thing we need is that
it's the dot product between the

457
00:35:00,000 --> 00:35:02,000
gradient and this vector u hat.
In particular,

458
00:35:02,000 --> 00:35:05,000
the directional derivatives in
the direction of I hat or j hat

459
00:35:05,000 --> 00:35:07,000
are just the usual partial
derivatives.

460
00:35:07,000 --> 00:35:12,000
That's what you would expect.
OK, and so now you see in

461
00:35:12,000 --> 00:35:15,000
particular if you try to go in a
direction that's perpendicular

462
00:35:15,000 --> 00:35:18,000
to the gradient,
then the directional derivative

463
00:35:18,000 --> 00:35:21,000
will be zero because you are
moving on the level curve.

464
00:35:21,000 --> 00:35:27,000
So, the value doesn't change,
OK?

465
00:35:27,000 --> 00:35:45,000
Questions about that?
Yes?

466
00:35:45,000 --> 00:35:49,000
Yeah, so let's see,
so indeed to look at more

467
00:35:49,000 --> 00:35:52,000
recent things,
if you are taking the flux

468
00:35:52,000 --> 00:35:55,000
through something given by an
equation,

469
00:35:55,000 --> 00:35:59,000
so, if you have a surface given
by an equation,

470
00:35:59,000 --> 00:36:05,000
say, f equals one.
So, say that you have a surface

471
00:36:05,000 --> 00:36:08,000
here or a curve given by an
equation,

472
00:36:08,000 --> 00:36:14,000
f equals constant,
then the normal vector to the

473
00:36:14,000 --> 00:36:19,000
surface is given by taking the
gradient of f.

474
00:36:19,000 --> 00:36:22,000
And that is,
in general, not a unit normal

475
00:36:22,000 --> 00:36:24,000
vector.
Now, if you wanted the unit

476
00:36:24,000 --> 00:36:28,000
normal vector to compute flux,
then you would just scale this

477
00:36:28,000 --> 00:36:30,000
guy down to unit length,
OK?

478
00:36:30,000 --> 00:36:33,000
So, if you wanted a unit
normal, that would be the

479
00:36:33,000 --> 00:36:37,000
gradient divided by its length.
However, for flux,

480
00:36:37,000 --> 00:36:40,000
that's still of limited
usefulness because you would

481
00:36:40,000 --> 00:36:42,000
still need to know about ds.
But, remember,

482
00:36:42,000 --> 00:36:46,000
we've seen a formula for flux
in terms of a non-unit normal

483
00:36:46,000 --> 00:36:52,000
vector, and n over n dot kdxdy.
So, indeed, this is how you

484
00:36:52,000 --> 00:36:58,000
could actually handle
calculations of flux through

485
00:36:58,000 --> 00:37:09,000
pretty much anything.
Any other questions about that?

486
00:37:09,000 --> 00:37:19,000
OK, so let me continue with a
couple more things we need to,

487
00:37:19,000 --> 00:37:25,000
so, we've seen how to do
min/max problems,

488
00:37:25,000 --> 00:37:33,000
in particular,
by looking at critical points.

489
00:37:33,000 --> 00:37:35,000
So, critical points,
remember, are the points where

490
00:37:35,000 --> 00:37:37,000
all the partial derivatives are
zero.

491
00:37:37,000 --> 00:37:40,000
So, if you prefer,
that's where the gradient

492
00:37:40,000 --> 00:37:45,000
vector is zero.
And, we know how to decide

493
00:37:45,000 --> 00:37:52,000
using the second derivative test
whether a critical point is

494
00:37:52,000 --> 00:37:57,000
going to be a local min,
a local max,

495
00:37:57,000 --> 00:38:02,000
or a saddle point.
Actually, we can't always quite

496
00:38:02,000 --> 00:38:05,000
decide because,
remember, we look at the second

497
00:38:05,000 --> 00:38:08,000
partials, and we compute this
quantity ac minus b squared.

498
00:38:08,000 --> 00:38:10,000
And, if it happens to be zero,
then actually we can't

499
00:38:10,000 --> 00:38:13,000
conclude.
But, most of the time we can

500
00:38:13,000 --> 00:38:16,000
conclude.
However, that's not all we need

501
00:38:16,000 --> 00:38:20,000
to look for an absolute global
maximum or minimum.

502
00:38:20,000 --> 00:38:23,000
For that, we also need to check
the boundary points,

503
00:38:23,000 --> 00:38:27,000
or look at the behavior of a
function, at infinity.

504
00:38:27,000 --> 00:38:38,000
So, we also need to check the
values of f at the boundary of

505
00:38:38,000 --> 00:38:46,000
its domain of definition or at
infinity.

506
00:38:46,000 --> 00:38:48,000
Just to give you an example
from single variable calculus,

507
00:38:48,000 --> 00:38:51,000
if you are trying to find the
minimum and the maximum of f of

508
00:38:51,000 --> 00:38:55,000
x equals x squared,
well, you'll find quickly that

509
00:38:55,000 --> 00:38:57,000
the minimum is at zero where x
squared is zero.

510
00:38:57,000 --> 00:39:00,000
If you are looking for the
maximum, you better not just

511
00:39:00,000 --> 00:39:02,000
look at the derivative because
you won't find it that way.

512
00:39:02,000 --> 00:39:05,000
However, if you think for a
second, you'll see that if x

513
00:39:05,000 --> 00:39:08,000
becomes very large,
then the function increases to

514
00:39:08,000 --> 00:39:10,000
infinity.
And, similarly,

515
00:39:10,000 --> 00:39:14,000
if you try to find the minimum
and the maximum of x squared

516
00:39:14,000 --> 00:39:17,000
when x varies only between one
and two,

517
00:39:17,000 --> 00:39:19,000
well, you won't find the
critical point,

518
00:39:19,000 --> 00:39:21,000
but you'll still find that the
smallest value of x squared is

519
00:39:21,000 --> 00:39:24,000
when x is at one,
and the largest is at x equals

520
00:39:24,000 --> 00:39:26,000
two.
And, all this business about

521
00:39:26,000 --> 00:39:29,000
boundaries and infinity is
exactly the same stuff,

522
00:39:29,000 --> 00:39:31,000
but with more than one
variable.

523
00:39:31,000 --> 00:39:37,000
It's just the story that maybe
the minimum and the maximum are

524
00:39:37,000 --> 00:39:42,000
not quite visible,
but they are at the edges of a

525
00:39:42,000 --> 00:39:48,000
domain we are looking at.
Well, in the last three

526
00:39:48,000 --> 00:39:55,000
minutes, I will just write down
a couple more things we've seen

527
00:39:55,000 --> 00:40:00,000
there.
So, how to do max/min problems

528
00:40:00,000 --> 00:40:08,000
with non-independent variables
-- So, if your variables are

529
00:40:08,000 --> 00:40:15,000
related by some condition,
g equals some constant.

530
00:40:15,000 --> 00:40:25,000
So, then we've seen the method
of Lagrange multipliers.

531
00:40:25,000 --> 00:40:31,000
OK, and what this method says
is that we should solve the

532
00:40:31,000 --> 00:40:36,000
equation gradient f equals some
unknown scalar lambda times the

533
00:40:36,000 --> 00:40:39,000
gradient, g.
So, that means each partial,

534
00:40:39,000 --> 00:40:43,000
f sub x equals lambda g sub x
and so on,

535
00:40:43,000 --> 00:40:48,000
and of course we have to keep
in mind the constraint equation

536
00:40:48,000 --> 00:40:53,000
so that we have the same number
of equations as the number of

537
00:40:53,000 --> 00:40:57,000
unknowns because you have a new
unknown here.

538
00:40:57,000 --> 00:41:04,000
And, the thing to remember is
that you have to be careful that

539
00:41:04,000 --> 00:41:13,000
the second derivative test does
not apply in this situation.

540
00:41:13,000 --> 00:41:16,000
I mean, this is only in the
case of independent variables.

541
00:41:16,000 --> 00:41:18,000
So, if you want to know if
something is a maximum or a

542
00:41:18,000 --> 00:41:20,000
minimum,
you just have to use common

543
00:41:20,000 --> 00:41:24,000
sense or compare the values of a
function at the various points

544
00:41:24,000 --> 00:41:29,000
you found.
Yes?

545
00:41:29,000 --> 00:41:34,000
Will we actually have to
calculate?

546
00:41:34,000 --> 00:41:38,000
Well, that depends on what the
problem asks you.

547
00:41:38,000 --> 00:41:40,000
It might ask you to just set up
the equations,

548
00:41:40,000 --> 00:41:41,000
or it might ask you to solve
them.

549
00:41:41,000 --> 00:41:44,000
So, in general,
solving might be difficult,

550
00:41:44,000 --> 00:41:47,000
but if it asks you to do it,
then it means it shouldn't be

551
00:41:47,000 --> 00:41:50,000
too hard.
I haven't written the final

552
00:41:50,000 --> 00:41:54,000
yet, so I don't know what it
will be, but it might be an easy

553
00:41:54,000 --> 00:42:00,000
one.
And, the last thing we've seen

554
00:42:00,000 --> 00:42:06,000
is constrained partial
derivatives.

555
00:42:06,000 --> 00:42:12,000
So, for example,
if you have a relation between

556
00:42:12,000 --> 00:42:15,000
x, y, and z,
which are constrained to be a

557
00:42:15,000 --> 00:42:20,000
constant,
then the notion of partial f

558
00:42:20,000 --> 00:42:24,000
partial x takes several
meanings.

559
00:42:24,000 --> 00:42:32,000
So, just to remind you very
quickly, there's the formal

560
00:42:32,000 --> 00:42:38,000
partial, partial f,
partial x, which means x

561
00:42:38,000 --> 00:42:43,000
varies.
Y and z are held constant.

562
00:42:43,000 --> 00:42:48,000
And, we forget the constraint.
This is not compatible with a

563
00:42:48,000 --> 00:42:51,000
constraint, but we don't care.
So, that's the guy that we

564
00:42:51,000 --> 00:42:54,000
compute just from the formula
for f ignoring the constraints.

565
00:42:54,000 --> 00:43:01,000
And then, we have the partial
f, partial x with y held

566
00:43:01,000 --> 00:43:06,000
constant, which means y held
constant.

567
00:43:06,000 --> 00:43:15,000
X varies, and now we treat z as
a dependent variable.

568
00:43:15,000 --> 00:43:20,000
It varies with x and y
according to whatever is needed

569
00:43:20,000 --> 00:43:24,000
so that this constraint keeps
holding.

570
00:43:24,000 --> 00:43:29,000
And, similarly,
there's partial f partial x

571
00:43:29,000 --> 00:43:33,000
with z held constant,
which means that,

572
00:43:33,000 --> 00:43:38,000
now, y is the dependent
variable.

573
00:43:38,000 --> 00:43:39,000
And, the way in which we
compute these,

574
00:43:39,000 --> 00:43:42,000
we've seen two methods which
I'm not going to tell you now

575
00:43:42,000 --> 00:43:45,000
because otherwise we'll be even
more over time.

576
00:43:45,000 --> 00:43:48,000
But, we've seen two methods for
computing these based on either

577
00:43:48,000 --> 00:43:50,000
the chain rule or on
differentials,

578
00:43:50,000 --> 00:43:52,000
solving and substituting into
differentials.