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Let's try to discuss a bit how
things relate to physics.

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There are two main things I
want to discuss.

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One of them is what curl says
about force fields and,

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in particular,a nice
consequence of that concerning

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gravitational attraction.
More about curl.

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If we have a velocity field,
then we have seen that the curl

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measures the rotation affects.
More precisely curl v measures

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twice the angular velocity,
or maybe I should say the

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angular velocity vector because
it also includes the axis of

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rotation.
I should say maybe for the

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rotation part of a motion.
For example,

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00:01:48,000 --> 00:01:58,000
just to remind you,
I mean we have seen this guy a

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couple of times,
but if I give you a uniform

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rotation motion about the z,
axes.

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00:02:14,000 --> 00:02:19,000
That is a vector field in which
the trajectories are going to be

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00:02:19,000 --> 00:02:23,000
circles centered in the z-axis
and our vector field is just

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going to be tangent to each of
these circles.

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00:02:26,000 --> 00:02:35,000
And, if you look at it from
above, then you will have this

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00:02:35,000 --> 00:02:42,000
rotation vector field that we
have seen many times.

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00:02:42,000 --> 00:02:48,000
Typically, the velocity vector
for this would be minus yi plus

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00:02:48,000 --> 00:02:53,000
yj times maybe a number that
represents how fast we are

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spinning,
the angular velocity in

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00:02:56,000 --> 00:03:01,000
gradients per second.
And then.

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00:03:01,000 --> 00:03:06,000
if you compute the curl of
this, you will end up with two

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00:03:06,000 --> 00:03:09,000
omega times k.
Now, the other kinds of vector

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00:03:09,000 --> 00:03:12,000
fields we have seen physically
are force fields.

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00:03:12,000 --> 00:03:17,000
The question is what does the
curl of a force field mean?

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What can we say about that?
The interpretation is a little

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bit less obvious,
but let's try to get some idea

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00:03:34,000 --> 00:03:42,000
of what it might be.
I want to remind you that if we

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00:03:42,000 --> 00:03:47,000
have a solid in a force field,
we can measure the torque

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00:03:47,000 --> 00:03:51,000
exerted by the force on the
solid.

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00:03:51,000 --> 00:03:55,000
Maybe first I should remind you
about what torque is in space.

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Let's say that I have a piece
of solid with a mass,

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delta m for example,
and I have a force that is

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being exerted to it.
Let's say that maybe my force

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might be F times delta m.
If you think,

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for example,
a gravitational field.

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The gravitational force is
actually the gravitational field

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00:04:20,000 --> 00:04:24,000
times the mass.
I mean you can forget delta m

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00:04:24,000 --> 00:04:29,000
if you don't like it.
And let's say that the position

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vector, which should be aiming
for the origin,

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00:04:33,000 --> 00:04:37,000
R is here.
And now let's say that maybe

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00:04:37,000 --> 00:04:42,000
this guy is at the end of some
arm or some metal thing and I

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want to hold it in place.
The force is going to exert a

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torque relative to the origin
that will try to measure how

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00:04:50,000 --> 00:04:54,000
much I am trying to swing this
guy around the origin.

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00:04:54,000 --> 00:04:57,000
And, consequently,
how much effort I have to exert

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if I want to actually maintain
its place by just holding it at

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the end of the stick here.
So the torque is now a vector,

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which is just the cross-product
of a position vector with a

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force.
What the torque measures again

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is the rotation effects of the
force.

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And if you remember the
principle that the derivative of

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velocity,
which is acceleration,

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is force divided by mass then
the derivative of angular

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velocity should be angular
acceleration which is related to

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the torque per unit mass.
To just remind you,

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00:06:04,000 --> 00:06:06,000
if I look at translation
motions,

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say I am just looking at the
point mass so there are no

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rotation effects then force
divided by mass is acceleration,

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00:06:21,000 --> 00:06:28,000
which is the derivative of
velocity.

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00:06:28,000 --> 00:06:35,000
And so what I am claiming is
that for rotation effects we

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have a similar law,
which maybe you have seen in

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8.01.
Well, it is one of the

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important things of solid
mechanics, which is the torque

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of a force divided by the moment
of inertia.

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I am cheating a little bit here.
If you can see how I am

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cheating then I am sure you know
how to state it correctly.

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And if you don't see how I am
cheating then let's just ignore

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the details.
[LAUGHTER]

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Is angular acceleration.
And angular acceleration is the

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derivative of angular velocity.
If I think of curl as an

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operation,
which from a velocity field

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gives the angular velocity of
its rotation effects,

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00:07:41,000 --> 00:07:46,000
then you see that the curl of
an acceleration field gives the

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00:07:46,000 --> 00:07:49,000
angular acceleration in the
rotation part of the

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acceleration effects.
And, therefore,

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the curl of a force field
measures the torque per unit

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moment of inertia.
It measures how much torque its

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force field exerts on a small
test solid placed in it.

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00:08:08,000 --> 00:08:12,000
If you have a small solid
somewhere, the curl will just

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00:08:12,000 --> 00:08:16,000
measure how much your solid
starts spinning if you leave it

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00:08:16,000 --> 00:08:18,000
in this force field.
In particular,

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a force field with no curl is a
force field that does not

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00:08:22,000 --> 00:08:26,000
generate any rotation motion.
That means if you put an object

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00:08:26,000 --> 00:08:29,000
in there that is completely
immobile and you leave it in

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00:08:29,000 --> 00:08:31,000
that force field,
well, of course it might

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accelerate in some direction but
it won't start spinning.

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While, if you put it in there
spinning already in some

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direction, it should continue to
spin in the same way.

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Of course, maybe there will be
friction and things like that

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00:08:50,000 --> 00:08:58,000
which will slow it down but this
force field is not responsible

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00:08:58,000 --> 00:09:05,000
for it.
The cool consequence of this is

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00:09:05,000 --> 00:09:14,000
if a force field F derives from
a potential -- That is what we

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have seen about conservative
forces.

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Our main concern so far has
been to say if we have a

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conservative force field it
means that the work of a force

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is the change in the energy.
And, in particular,

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we cannot get energy for free
out of it.

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And the change in the potential
energy is going to be the change

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in kinetic energy.
You have conservation of energy

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principles.
There is another thing that we

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know now because if a force
derives from a potential then

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that means its curl is zero.
That is the criterion we have

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seen for a vector field to
derive from a potential.

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00:09:58,000 --> 00:10:14,000
And if the curl is zero then it
means that this force does not

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00:10:14,000 --> 00:10:23,000
generate any rotation effects.
For example,

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00:10:23,000 --> 00:10:27,000
if you try to understand where
the earth comes from,

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well, the earth is spinning on
itself as it goes around the

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sun.
And you might wonder where that

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comes from.
Is that causes by gravitational

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attraction?
And the answer is no.

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Gravitational attraction in
itself cannot cause the earth to

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00:10:44,000 --> 00:10:47,000
start spinning faster or slower,
at least if you assume the

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earth to be a solid,
which actually is false.

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I mean basically the reason why
the earth is spinning is because

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it was formed spinning.
It didn't start spinning

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00:11:01,000 --> 00:11:03,000
because of gravitational
effects.

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00:11:03,000 --> 00:11:08,000
And that is a rather deep
purely mathematical consequence

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00:11:08,000 --> 00:11:12,000
of understanding gravitation in
this way.

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00:11:12,000 --> 00:11:16,000
It is quite spectacular that
just by abstract thinking we got

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00:11:16,000 --> 00:11:17,000
there.
What is the truth?

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00:11:17,000 --> 00:11:21,000
Well, the truth is the earth,
the moon and everything is

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slightly deformable.
And so there is deformation,

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friction effects,
tidal effects and so on.

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And these actually cause
rotations to get slightly

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synchronized with each other.
For example,

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00:11:32,000 --> 00:11:36,000
if you want to explain why the
moon is always showing the same

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00:11:36,000 --> 00:11:39,000
face to the earth,
why the rotation of a moon on

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00:11:39,000 --> 00:11:43,000
itself is synchronized with its
revolution around the earth,

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00:11:43,000 --> 00:11:47,000
which is actually explained by
friction effects over time and

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00:11:47,000 --> 00:11:50,000
the gravitational attraction of
the earth and the moon.

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There is something there,
but if you took perfectly

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00:11:59,000 --> 00:12:09,000
rigid, solid bodies then
gravitation would never cause

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00:12:09,000 --> 00:12:15,000
any rotation effects.
Of course that tells us that we

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00:12:15,000 --> 00:12:20,000
do not know how to answer the
question of why is the earth

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00:12:20,000 --> 00:12:22,000
spinning.
That will be left for another

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physics class.
I don't have a good answer to

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00:12:31,000 --> 00:12:35,000
that.
That was kind of 8.01-ish.

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00:12:35,000 --> 00:12:40,000
Let me now move forward to 8.02
stuff.

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I want to tell you things about
electric and magnetic fields.

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00:12:54,000 --> 00:13:01,000
And, in fact,
something that is known as

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Maxwell's equations.
Just a quick poll.

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00:13:06,000 --> 00:13:10,000
How many of you have been
taking 8.02 or something like

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00:13:10,000 --> 00:13:13,000
that?
OK. That is not very many.

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00:13:13,000 --> 00:13:15,000
For most of you this is a
preview.

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00:13:15,000 --> 00:13:18,000
If you have been taking 8.02,
have you seen Maxwell's

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equations, at least part of
them?

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Yeah.
OK.

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Then I am sure,
in that case,

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00:13:23,000 --> 00:13:25,000
you know better than me what I
am going to talk about because I

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00:13:25,000 --> 00:13:30,000
am not a physicist.
But just in case.

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00:13:30,000 --> 00:13:35,000
Maxwell's equations govern how
electric and magnetic fields

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00:13:35,000 --> 00:13:39,000
behave, how they are caused by
electric charges and their

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00:13:39,000 --> 00:13:41,000
motions.
And, in particular,

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00:13:41,000 --> 00:13:45,000
they explain a lot of things
such as how electric devices

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00:13:45,000 --> 00:13:49,000
work, but also how
electromagnetic waves propagate.

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00:13:49,000 --> 00:13:54,000
In particular,
that explains light and all

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00:13:54,000 --> 00:13:58,000
sorts of waves.
It is thanks to them,

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00:13:58,000 --> 00:14:02,000
you know, your cell phone,
laptops and things like that

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00:14:02,000 --> 00:14:06,000
work.
Anyway.

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00:14:06,000 --> 00:14:11,000
Hopefully most of you know that
the electric field is a vector

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00:14:11,000 --> 00:14:14,000
field that basically tells you
what kind of force will be

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exerted on a charged particle
that you put in it.

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00:14:18,000 --> 00:14:23,000
If you have a particle carrying
an electric charge then this

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00:14:23,000 --> 00:14:27,000
vector field will tell you,
basically there will be an

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00:14:27,000 --> 00:14:31,000
electric force which is the
charge times E that will be

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00:14:31,000 --> 00:14:33,000
exerted on that particle.
And that is what is

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00:14:33,000 --> 00:14:36,000
responsible, for example,
for the flow of electrons when

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00:14:36,000 --> 00:14:41,000
you have a voltage difference.
Because classically this guy is

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00:14:41,000 --> 00:14:45,000
a gradient of a potential.
And that potential is just

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00:14:45,000 --> 00:14:50,000
electric voltage.
The magnetic field is a little

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00:14:50,000 --> 00:14:55,000
bit harder to think about if you
have never seen it in physics,

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00:14:55,000 --> 00:15:00,000
but it is what is causing,
for example,

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00:15:00,000 --> 00:15:04,000
magnets to work.
Well, basically it is a force

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00:15:04,000 --> 00:15:09,000
that is also expressed in terms
of a vector field usually called

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B.
Some people call it H but I am

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going to use B.
And that force tends to cause

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it, if you have a moving charged
particle, to deflect its

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00:15:20,000 --> 00:15:24,000
trajectory and start rotating in
a magnetic field.

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What it does is not quite as
easy as what an electric field

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does.
Just to give you formulas,

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00:15:35,000 --> 00:15:39,000
the force caused by the
electric field is the charge

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00:15:39,000 --> 00:15:43,000
times the electric field.
And the force caused by the

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00:15:43,000 --> 00:15:47,000
magnetic field,
I am never sure about the sign.

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00:15:47,000 --> 00:15:52,000
Is that the correct sign?
Good.

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00:15:52,000 --> 00:15:56,000
Now, the question is we need to
understand how these fields

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themselves are caused by the
charged particles that are

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placed in them.
There are various laws in there

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that explain what is going on.
Let me focus today on the

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electric field.
Maxwell's equations actually

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tell you about div and curl of
these fields.

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00:16:22,000 --> 00:16:27,000
Let's look at div and curl of
the electric field.

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00:16:27,000 --> 00:16:37,000
The first equation is called
the Gauss-Coulomb law.

202
00:16:37,000 --> 00:16:47,000
And it says that the divergence
of the electric field is equal

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to,
so this is a just a physical

204
00:16:51,000 --> 00:16:54,000
constant,
and what it is equal to depends

205
00:16:54,000 --> 00:16:57,000
on what units you are using.
And this guy rho,

206
00:16:57,000 --> 00:17:01,000
well, it is not the same rho as
in spherical coordinates because

207
00:17:01,000 --> 00:17:06,000
physicists somehow pretended
they used that letter first.

208
00:17:06,000 --> 00:17:08,000
It is the electric charge
density.

209
00:17:08,000 --> 00:17:15,000
It is the amount of electric
charge per unit volume.

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00:17:15,000 --> 00:17:20,000
What this tells you is that
divergence of E is caused by the

211
00:17:20,000 --> 00:17:23,000
presence of electric charge.
In particular,

212
00:17:23,000 --> 00:17:29,000
if you have an empty region of
space or a region where nothing

213
00:17:29,000 --> 00:17:34,000
has electrical charge then E has
divergence equal to zero.

214
00:17:34,000 --> 00:17:38,000
Now, that looks like a very
abstract strange equation.

215
00:17:38,000 --> 00:17:43,000
I mean it is a partial
differential equation satisfied

216
00:17:43,000 --> 00:17:49,000
by the electric field E.
And that is not very intuitive

217
00:17:49,000 --> 00:17:56,000
in any way.
What is actually more intuitive

218
00:17:56,000 --> 00:18:05,000
is what we get if we apply the
divergence theorem to this

219
00:18:05,000 --> 00:18:11,000
equation.
If I think now about any closed

220
00:18:11,000 --> 00:18:16,000
surface,
and I want to think about the

221
00:18:16,000 --> 00:18:21,000
flux of the electric field out
of that surface,

222
00:18:21,000 --> 00:18:24,000
we haven't really thought about
what the flux of a force field

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00:18:24,000 --> 00:18:27,000
does.
And I don't want to get into

224
00:18:27,000 --> 00:18:31,000
that because there is no very
easy answer in general,

225
00:18:31,000 --> 00:18:35,000
but I am going to explain soon
how this can be useful

226
00:18:35,000 --> 00:18:38,000
sometimes.
Let's say that we want to find

227
00:18:38,000 --> 00:18:43,000
the flux of the electric field
out of a closed surface.

228
00:18:43,000 --> 00:18:47,000
Then, by the divergence
theorem,

229
00:18:47,000 --> 00:18:53,000
that is equal to the triple
integral of a region inside of

230
00:18:53,000 --> 00:18:57,000
div E dV,
which is by the equation one

231
00:18:57,000 --> 00:19:00,000
over epsilon zero,
that is this constant,

232
00:19:00,000 --> 00:19:06,000
times the triple integral of
rho dV.

233
00:19:06,000 --> 00:19:09,000
But now, if I integrate the
charge density over the entire

234
00:19:09,000 --> 00:19:12,000
region,
then what I will get is

235
00:19:12,000 --> 00:19:17,000
actually the total amount of
electric charge inside the

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00:19:17,000 --> 00:19:28,000
region.
That is the electric charge in

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00:19:28,000 --> 00:19:31,000
D.
This one tells us,

238
00:19:31,000 --> 00:19:34,000
in a more concrete way,
how electric charges placed in

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00:19:34,000 --> 00:19:38,000
here influence the electric
field around them.

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00:19:38,000 --> 00:19:40,000
In particular,
one application of that is if

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00:19:40,000 --> 00:19:43,000
you want to study capacitors.
Capacitors are these things

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00:19:43,000 --> 00:19:46,000
that store energy by basically
you have two plates,

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00:19:46,000 --> 00:19:49,000
one that contains positive
charge and a negative charge.

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00:19:49,000 --> 00:19:52,000
Then you have a voltage between
these plates.

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00:19:52,000 --> 00:19:57,000
And, basically,
that can provide electrical

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00:19:57,000 --> 00:20:03,000
energy to power maybe an
electric circuit.

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00:20:03,000 --> 00:20:06,000
That is not really a battery
because it doesn't store energy

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00:20:06,000 --> 00:20:08,000
in large enough amounts.
But, for example,

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00:20:08,000 --> 00:20:11,000
that is why when you switch
your favorite gadget off it

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00:20:11,000 --> 00:20:14,000
doesn't actually go off
immediately but somehow you see

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00:20:14,000 --> 00:20:18,000
things dimming progressively.
There is a capacitor in there.

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00:20:18,000 --> 00:20:20,000
If you want to understand how
the voltage and the charge

253
00:20:20,000 --> 00:20:23,000
relate to each other,
the voltage is obtained by

254
00:20:23,000 --> 00:20:26,000
integrating the electric field
from one plate to the other

255
00:20:26,000 --> 00:20:29,000
plate.
And the charges in the plates

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00:20:29,000 --> 00:20:34,000
are what causes the electric
field between the plates.

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00:20:34,000 --> 00:20:37,000
That is how you can get the
relation between voltage and

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00:20:37,000 --> 00:20:41,000
charge in these guys.
That is an example of

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00:20:41,000 --> 00:20:44,000
application of that.
Now, of course,

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if you haven't seen any of this
then maybe it is a little bit

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00:20:49,000 --> 00:20:54,000
esoteric, but that will tell you
part of what you will see in

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00:20:54,000 --> 00:20:59,000
8.02.
Questions?

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00:20:59,000 --> 00:21:07,000
I see some confused faces.
Well, don't worry.

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It will make sense some day.
[LAUGHTER]

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The next one I want to tell you
about is Faraday's law.

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00:21:23,000 --> 00:21:25,000
In case you are confused,
Maxwell's equations,

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00:21:25,000 --> 00:21:29,000
there are four equations in the
set of Maxwell's equations and

268
00:21:29,000 --> 00:21:31,000
most of them don't carry
Maxwell's name.

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00:21:31,000 --> 00:21:40,000
That is a quirky feature.
That one tells you about the

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00:21:40,000 --> 00:21:44,000
curl of the electric field.
Now, depending on your

271
00:21:44,000 --> 00:21:46,000
knowledge,
you might start telling me that

272
00:21:46,000 --> 00:21:50,000
the curl of the electric field
has to be zero because it is the

273
00:21:50,000 --> 00:21:52,000
gradient of the electric
potential.

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00:21:52,000 --> 00:21:54,000
I told you this stuff about
voltage.

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00:21:54,000 --> 00:21:58,000
Well, that doesn't account for
the fact that sometimes you can

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00:21:58,000 --> 00:22:02,000
create voltage out of nowhere
using magnetic fields.

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00:22:02,000 --> 00:22:05,000
And, in fact,
you have a failure of

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00:22:05,000 --> 00:22:09,000
conservativity of the electric
force if you have a magnetic

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00:22:09,000 --> 00:22:12,000
field.
What this one says is the curl

280
00:22:12,000 --> 00:22:17,000
of E is not zero but rather it
is the derivative of the

281
00:22:17,000 --> 00:22:21,000
magnetic field with respect to
time.

282
00:22:21,000 --> 00:22:26,000
More precisely it tells you
that what you might have learned

283
00:22:26,000 --> 00:22:31,000
about electric fields deriving
from electric potential becomes

284
00:22:31,000 --> 00:22:35,000
false if you have a variable
magnetic field.

285
00:22:35,000 --> 00:22:41,000
And just to tell you again that
is a strange partial

286
00:22:41,000 --> 00:22:47,000
differential equation relating
these two vector fields.

287
00:22:47,000 --> 00:22:51,000
To make sense of it one should
use Stokes' theorem.

288
00:22:51,000 --> 00:22:56,000
If we apply Stokes' theorem to
compute the work done by the

289
00:22:56,000 --> 00:23:00,000
electric field around a closed
curve,

290
00:23:00,000 --> 00:23:04,000
that means you have a wire in
there and you want to find the

291
00:23:04,000 --> 00:23:07,000
voltage along the wire.
Now there is a strange thing

292
00:23:07,000 --> 00:23:10,000
because classically you would
say, well, if I just have a wire

293
00:23:10,000 --> 00:23:13,000
with nothing in it there is no
voltage on it.

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00:23:13,000 --> 00:23:18,000
Well, a small change in plans.
If you actually have a varying

295
00:23:18,000 --> 00:23:23,000
magnetic field that passes
through that wire then that will

296
00:23:23,000 --> 00:23:31,000
actually generate voltage in it.
That is how a transformer works.

297
00:23:31,000 --> 00:23:34,000
When you plug your laptop into
the wall circuit,

298
00:23:34,000 --> 00:23:36,000
you don't actually feed it
directly 110 volts,

299
00:23:36,000 --> 00:23:40,000
120 volts or whatever.
There is a transformer in there.

300
00:23:40,000 --> 00:23:45,000
What the transformer does it
takes some input voltage and

301
00:23:45,000 --> 00:23:49,000
passes that through basically a
loop of wire.

302
00:23:49,000 --> 00:23:53,000
Not much seems to be happening.
But now you have another loops

303
00:23:53,000 --> 00:23:56,000
of wire that is intertwined with
it.

304
00:23:56,000 --> 00:23:59,000
Somehow the magnetic field
generated by it,

305
00:23:59,000 --> 00:24:03,000
and it has to be a donating
current.

306
00:24:03,000 --> 00:24:06,000
The donating current varies
over time in the first wire.

307
00:24:06,000 --> 00:24:09,000
That generates a magnetic field
that varies over time,

308
00:24:09,000 --> 00:24:13,000
so that causes 2B by 2t and
that causes curl of the electric

309
00:24:13,000 --> 00:24:15,000
field.
And the curl of the electric

310
00:24:15,000 --> 00:24:18,000
field will generate voltage
between these two guys.

311
00:24:18,000 --> 00:24:21,000
And that is how a transformer
works.

312
00:24:21,000 --> 00:24:25,000
It uses Stokes' theorem.
More precisely,

313
00:24:25,000 --> 00:24:28,000
how do we find the voltage
between these two points?

314
00:24:28,000 --> 00:24:32,000
Well, let's close the loop and
let's try to figure out the

315
00:24:32,000 --> 00:24:37,000
voltage inside this loop.
To find a voltage along a

316
00:24:37,000 --> 00:24:42,000
closed curve places in a varying
magnetic field,

317
00:24:42,000 --> 00:24:47,000
we have to do the line integral
along a closed curve of the

318
00:24:47,000 --> 00:24:51,000
electric field.
And you should think of this as

319
00:24:51,000 --> 00:24:54,000
the voltage generated in this
circuit.

320
00:24:54,000 --> 00:25:05,000
That will be the flux for this
surface bounded by the curve of

321
00:25:05,000 --> 00:25:11,000
curl E dot dS.
That is what Stokes' theorem

322
00:25:11,000 --> 00:25:14,000
says.
And now if you combine that

323
00:25:14,000 --> 00:25:21,000
with Faraday's law you end up
with the flux trough S of minus

324
00:25:21,000 --> 00:25:25,000
dB over dt.
And, of course, you could take,

325
00:25:25,000 --> 00:25:27,000
if your loop doesn't move over
time,

326
00:25:27,000 --> 00:25:31,000
I mean there is a different
story if you start somehow

327
00:25:31,000 --> 00:25:34,000
taking your wire and somehow
moving it inside the field.

328
00:25:34,000 --> 00:25:37,000
But if you don't do that,
if it is the field that is

329
00:25:37,000 --> 00:25:40,000
moving then you just can take
the dB by dt outside.

330
00:25:40,000 --> 00:25:48,000
But let's not bother.
Again, what this equation tells

331
00:25:48,000 --> 00:25:52,000
you is that if the magnetic
field changes over time then it

332
00:25:52,000 --> 00:25:55,000
creates, just out of nowhere,
and electric field.

333
00:25:55,000 --> 00:26:09,000
And that electric field can be
used to power up things.

334
00:26:09,000 --> 00:26:11,000
I don't really claim that I
have given you enough details to

335
00:26:11,000 --> 00:26:15,000
understand how they work,
but basically these equations

336
00:26:15,000 --> 00:26:20,000
are the heart of understanding
how things like capacitors and

337
00:26:20,000 --> 00:26:23,000
transformers work.
And they also explain a lot of

338
00:26:23,000 --> 00:26:25,000
other things,
but I will leave that to your

339
00:26:25,000 --> 00:26:28,000
physics teachers.
Just for completeness,

340
00:26:28,000 --> 00:26:33,000
I will just give you the last
two equations in that.

341
00:26:33,000 --> 00:26:37,000
I am not even going to try to
explain them too much.

342
00:26:37,000 --> 00:26:42,000
One of them says that the
divergence of the magnetic field

343
00:26:42,000 --> 00:26:45,000
is zero,
which somehow is fortunate

344
00:26:45,000 --> 00:26:49,000
because otherwise you would run
into trouble trying to

345
00:26:49,000 --> 00:26:53,000
understand surface independence
when you apply Stokes' theorem

346
00:26:53,000 --> 00:26:58,000
in here.
And the last one tells you how

347
00:26:58,000 --> 00:27:04,000
the curl of the magnetic field
is caused by motion of charged

348
00:27:04,000 --> 00:27:09,000
particles.
In fact, let's say that the

349
00:27:09,000 --> 00:27:16,000
curl of B is given by this kind
of formula, well,

350
00:27:16,000 --> 00:27:23,000
J is what is called the vector
of current density.

351
00:27:23,000 --> 00:27:30,000
It measures the flow of
electrically charged particles.

352
00:27:30,000 --> 00:27:34,000
You get this guy when you start
taking charged particles,

353
00:27:34,000 --> 00:27:38,000
like electrons maybe,
and moving them around.

354
00:27:38,000 --> 00:27:40,000
And, of course,
that is actually part of how

355
00:27:40,000 --> 00:27:44,000
transformers work because I have
told you running the AC through

356
00:27:44,000 --> 00:27:46,000
the first loop generates a
magnetic field.

357
00:27:46,000 --> 00:27:49,000
Well, how does it do that?
It is thanks to this equation.

358
00:27:49,000 --> 00:27:52,000
If you have a current passing
in the loop that causes a

359
00:27:52,000 --> 00:27:54,000
magnetic field and,
in turn, for the other equation

360
00:27:54,000 --> 00:27:59,000
that causes an electric field,
which in turn causes a current.

361
00:27:59,000 --> 00:28:08,000
It is all somehow intertwined
in a very intricate way and is

362
00:28:08,000 --> 00:28:15,000
really remarkable how well that
works in practice.

363
00:28:15,000 --> 00:28:17,000
I think that is basically all I
wanted to say about 8.02.

364
00:28:17,000 --> 00:28:23,000
I don't want to put your
physics teachers out of a job.

365
00:28:23,000 --> 00:28:24,000
[LAUGHTER]
If you haven't seen any of this

366
00:28:24,000 --> 00:28:26,000
before,
I understand that this is

367
00:28:26,000 --> 00:28:28,000
probably not detailed enough to
be really understandable,

368
00:28:28,000 --> 00:28:32,000
but hopefully it will make you
a bit curious about that and

369
00:28:32,000 --> 00:28:36,000
prompt you to take that class
someday and maybe even remember

370
00:28:36,000 --> 00:28:39,000
how it relates to 18.02.