Partial derivative of a parametric surface, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.352
- [Voiceover] So we've just
computed a vector-valued

00:00:02.352 --> 00:00:05.564
partial derivative of a
vector-valued function,

00:00:05.564 --> 00:00:06.940
but the question is What does this mean?

00:00:06.940 --> 00:00:09.211
What does this jumble
of symbols actually mean

00:00:09.211 --> 00:00:12.422
in a more intuitive geometric setting?

00:00:12.422 --> 00:00:13.755
And that has everything to do with

00:00:13.755 --> 00:00:15.410
how you visualize the function,

00:00:15.410 --> 00:00:16.961
and with this specific function,

00:00:16.961 --> 00:00:19.545
given that the input is two-dimensional

00:00:19.545 --> 00:00:21.233
but the output is three-dimensional,

00:00:21.233 --> 00:00:23.648
meaning the output has more
dimensions than the input,

00:00:23.648 --> 00:00:26.802
it's nice to visualize it
as a parametric surface.

00:00:26.802 --> 00:00:28.097
And the way that I do that,

00:00:28.097 --> 00:00:29.590
maybe you could call this visualizing it

00:00:29.590 --> 00:00:31.264
as a transformation also

00:00:31.264 --> 00:00:35.253
'cause what I want to do is
basically think of the TS plane,

00:00:35.253 --> 00:00:37.003
think of the TS plane

00:00:38.334 --> 00:00:40.226
where all these input values live

00:00:40.226 --> 00:00:42.350
and kind of think of how that's gonna map

00:00:42.350 --> 00:00:44.347
into three-dimensional space,

00:00:44.347 --> 00:00:47.131
but when I do that, I'm actually
gonna cheat a little bit.

00:00:47.131 --> 00:00:48.949
Rather than having a
separate plane off there

00:00:48.949 --> 00:00:52.357
as the TS plane, I'm
going to kind of overwrite

00:00:52.357 --> 00:00:54.175
onto the XY plane itself

00:00:54.175 --> 00:00:57.212
and plop the TS plane down like this,

00:00:57.212 --> 00:00:58.473
and this isn't the full TS plane.

00:00:58.473 --> 00:01:00.976
This is actually supposed to represent

00:01:00.976 --> 00:01:05.155
just values of T that range
from zero up to three,

00:01:05.155 --> 00:01:08.035
so each tick mark on the graph
here corresponds with a half.

00:01:08.035 --> 00:01:12.348
So this is one, that's two,
and then up here is three.

00:01:12.348 --> 00:01:13.181
Same with S.

00:01:13.181 --> 00:01:15.413
S also ranges from zero to three,

00:01:15.413 --> 00:01:16.430
and the reason that I'm plopping it

00:01:16.430 --> 00:01:18.178
inside three-dimensional
space to start with,

00:01:18.178 --> 00:01:21.436
kind of overwriting the XY
plane with the TS plane,

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is just to make the animating
a little bit easier.

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You could call it laziness.

00:01:26.999 --> 00:01:28.944
But the benefit here is what we can do

00:01:28.944 --> 00:01:31.888
is watch each point, and
each one of these points

00:01:31.888 --> 00:01:35.808
you're thinking of is corresponding
to some kind of TS pair,

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an input point, which is
just a pair of numbers,

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and we're gonna watch
each one of those points

00:01:40.296 --> 00:01:42.358
move to the corresponding output.

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The output is a three-dimensional value,

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a three-dimensional vector or point,

00:01:45.966 --> 00:01:47.846
however you wanna think about it,

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and what that looks like when
we animate this actually is

00:01:53.048 --> 00:01:56.609
each one of those points
in our square of TS plane

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moves to the corresponding output,

00:01:59.792 --> 00:02:01.484
and you end up with a certain surface.

00:02:01.484 --> 00:02:02.655
And just to make it a
little more concrete,

00:02:02.655 --> 00:02:04.409
what's actually going on here,

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let's focus in on just one point,

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and we'll focus in on this point

00:02:07.922 --> 00:02:09.752
not just for the function visualization

00:02:09.752 --> 00:02:12.069
but for the partial derivative as well,

00:02:12.069 --> 00:02:14.386
and the function, or the point rather,

00:02:14.386 --> 00:02:17.732
that I care about is gonna be at one, one.

00:02:17.732 --> 00:02:20.360
So this point right here represents

00:02:20.360 --> 00:02:21.527
the pair of TS

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where each one of them is equal to one.

00:02:26.078 --> 00:02:26.911
One, one.

00:02:28.017 --> 00:02:30.067
And you can start by
predicting where you think

00:02:30.067 --> 00:02:31.711
this is gonna get output,

00:02:31.711 --> 00:02:34.260
and to do that, you just
plug it into the function.

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This is kind of what
the visualization means

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as we're plugging this in,

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and for T and S, we're gonna
plug in just one and one.

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So that top part is gonna
look like one squared

00:02:43.439 --> 00:02:46.457
minus one squared, which becomes zero.

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That middle part is gonna be
one times one, which is one,

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and then over here we're gonna
have one times one squared,

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which is one minus one
times one times one squared,

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which is, again, one.

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And you can probably see,
'cause of the symmetry there,

00:02:57.530 --> 00:03:00.230
those also cancel out, you get zero,

00:03:00.230 --> 00:03:03.505
which means the output
corresponding with this input

00:03:03.505 --> 00:03:05.433
should be the vector zero, one, zero,

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a vector that's of unit length
pointing in the Y direction.

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So if we look here, this is the X axis.

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This here is the Y axis.

00:03:12.349 --> 00:03:15.352
So you would think it should be a vector

00:03:15.352 --> 00:03:17.907
that looks kind of like this.

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Unit vector in the Y direction,

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and the point of the
surface is what corresponds

00:03:20.848 --> 00:03:22.744
to the tip of that vector.

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This is how we visualize
parametric things.

00:03:24.473 --> 00:03:26.273
You just think of the tip of the vector

00:03:26.273 --> 00:03:28.801
as kinda moving through space
and drawing out the thing.

00:03:28.801 --> 00:03:31.143
In this case, the thing
it's drawing is a surface.

00:03:31.143 --> 00:03:32.669
So if we watch that animation again

00:03:32.669 --> 00:03:34.986
and we let things play forward,

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that dot corresponding
with the input one, one

00:03:38.251 --> 00:03:41.160
does indeed land at
the tip of that vector.

00:03:41.160 --> 00:03:42.479
So, at least for that value,

00:03:42.479 --> 00:03:46.384
you can see that I'm not lying
to you with the animation.

00:03:46.384 --> 00:03:48.780
And in principle, you could do
that for every single point.

00:03:48.780 --> 00:03:51.933
If any given input point, you
plug it through the function,

00:03:51.933 --> 00:03:54.232
and you draw the vector in
three-dimensional space,

00:03:54.232 --> 00:03:55.440
as you watch this animation,

00:03:55.440 --> 00:03:58.320
it'll land at the tip of that vector.

00:03:58.320 --> 00:03:59.153
So...

00:04:00.976 --> 00:04:02.078
Now, if we want to start thinking about

00:04:02.078 --> 00:04:04.366
what the partial derivative means,

00:04:04.366 --> 00:04:07.212
remember this little DT, this partial T,

00:04:07.212 --> 00:04:09.894
is telling you to nudge
it in the T direction.

00:04:09.894 --> 00:04:11.389
So what does movement,

00:04:11.389 --> 00:04:13.525
not even nudges but just
movement in general,

00:04:13.525 --> 00:04:17.892
look like in the T direction
for our little snippet

00:04:17.892 --> 00:04:20.070
of the TS plane here?

00:04:20.070 --> 00:04:21.903
Well, the T direction,

00:04:23.181 --> 00:04:24.755
I'm saying is in this direction here

00:04:24.755 --> 00:04:29.047
where this represents one,
two, three of T values,

00:04:29.047 --> 00:04:32.886
and this line here represents
the constant value for S,

00:04:32.886 --> 00:04:35.393
so this would be S
constantly equaling one,

00:04:35.393 --> 00:04:37.332
which you can know because
it's passing through

00:04:37.332 --> 00:04:39.246
the point one, one.

00:04:39.246 --> 00:04:41.816
And then otherwise you're
just letting T range freely.

00:04:41.816 --> 00:04:43.936
And if we watch how this gets transformed

00:04:43.936 --> 00:04:45.890
under the transformation,

00:04:45.890 --> 00:04:48.601
under the mapping to
the parametric surface,

00:04:48.601 --> 00:04:51.804
you can get a feel for
what varying the input T

00:04:51.804 --> 00:04:53.887
does in the output space.

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So this whole pink line
is basically telling you

00:04:57.325 --> 00:05:00.571
what happens if you let
S constantly equal one

00:05:00.571 --> 00:05:04.288
but you let the variable T,
the input T, vary freely,

00:05:04.288 --> 00:05:07.562
and you get a certain curve
in three-dimensional space.

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And if you had a different constant for S,

00:05:10.042 --> 00:05:11.013
it would be another curve,

00:05:11.013 --> 00:05:12.993
and maybe you can kinda
see on the grid lines

00:05:12.993 --> 00:05:14.451
what shape those other curves would have,

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and they're all, in a sense, parallelish

00:05:17.407 --> 00:05:21.107
to this curve corresponding
to S equals one.

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So if, instead of thinking
about movement of T as whole,

00:05:24.129 --> 00:05:26.964
you start thinking about nudges,

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this whole partial T

00:05:29.962 --> 00:05:31.176
is something where we're just imagining

00:05:31.176 --> 00:05:33.849
a tiny, tiny, little
movement in the T direction.

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Not really that much,
just a tiny, little move.

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It's like you're recording
its value as partial T,

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so maybe, if you're being concrete,

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you'd say partial T would
be something like 0.01.

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And really it's gonna
be a limiting variable

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that gets smaller and smaller,

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but I find it's kind
of nice to think about

00:05:48.131 --> 00:05:50.376
an actual value like one one-hundredth.

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And then if you let this whole thing

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undergo the transformation

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and we kind of watch the input point,

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watch the line representing T,

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that little nudge, that little nudge

00:06:00.015 --> 00:06:01.953
is gonna get maybe stretched or squished,

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and it's gonna result
in some kind of vector

00:06:05.035 --> 00:06:06.551
pointing along that curve,

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and it'll be tangent to that curve.

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The vector that tells you how you move

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just a tiny, tiny little bit

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will be tangent in some way,

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and that vector, that output nudge,

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is what you're thinking
of as your tiny change

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to the output vector, that partial V.

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And when you divide it by the tiny value,

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if your tiny value was 0.01
and you divide it by that,

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it's gonna become something bigger,

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so the actual derivative isn't gonna be

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just some tiny, little nudge

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that's hardly, hardly visible,

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but it's gonna be that nudge
vector scaled appropriately.

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In this case, it would be
divided by one one-hundredth

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and multiplied by 100,

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and it would be something that
remains tangent to the curve,

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but maybe it's pointing big.

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And the larger it is, the longer it is,

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that's telling you that as you let T vary

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and you're kind of moving
along this pink curve,

00:06:56.964 --> 00:06:59.745
tiny nudges in T correspond
with larger movements.

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The ratio of the nudge size is bigger.

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So if you were to have a very
long partial derivative vector

00:07:06.216 --> 00:07:08.069
that's still tangent but
really goes out there,

00:07:08.069 --> 00:07:09.422
that would tell you that as you vary T

00:07:09.422 --> 00:07:12.368
you're zipping along super quickly.

00:07:12.368 --> 00:07:15.435
And if we just look at this particular one

00:07:15.435 --> 00:07:16.631
that's zooming off of one,

00:07:16.631 --> 00:07:18.669
you kind of get a feel for
the curve around that point.

00:07:18.669 --> 00:07:19.502
You say, "Okay, okay."

00:07:19.502 --> 00:07:22.108
And that curve, you're moving
positively in the X direction.

00:07:22.108 --> 00:07:23.998
You're moving to the right.

00:07:23.998 --> 00:07:26.493
You're moving positively
in the Z direction.

00:07:26.493 --> 00:07:27.813
Not Z, sorry, the Y.

00:07:27.813 --> 00:07:31.089
Positively in the Y direction up there.

00:07:31.089 --> 00:07:32.602
And the Z direction is
actually negative, isn't it?

00:07:32.602 --> 00:07:35.802
This curve kind of goes down
as far as Z is concerned.

00:07:35.802 --> 00:07:37.504
So before even computing it,

00:07:37.504 --> 00:07:38.915
if I were to tell you
that I'm gonna plug in

00:07:38.915 --> 00:07:41.412
the value one, one to
this partial derivative

00:07:41.412 --> 00:07:43.880
that we computed in the last video,

00:07:43.880 --> 00:07:46.562
you would say, "Oh, well,
just looking at the picture,

00:07:46.562 --> 00:07:47.985
"you can kind of tell that the X value

00:07:47.985 --> 00:07:49.408
"is gonna be something positive,

00:07:49.408 --> 00:07:51.068
something greater than zero."

00:07:51.068 --> 00:07:53.448
The Y value is also gonna
be something positive,

00:07:53.448 --> 00:07:55.807
and, again, that's because
the movement is to the right,

00:07:55.807 --> 00:07:56.859
so positive X.

00:07:56.859 --> 00:07:58.649
It's moving up, so positive Y,

00:07:58.649 --> 00:08:01.031
but the Z value should actually
be a little bit negative

00:08:01.031 --> 00:08:03.075
because as you look at this curve,

00:08:03.075 --> 00:08:05.919
it's going down in a sense.

00:08:05.919 --> 00:08:07.327
And with that being our prediction,

00:08:07.327 --> 00:08:11.077
if you start plugging
in one, one to T and S,

00:08:12.035 --> 00:08:16.585
what you'll see is that
two times one is two,

00:08:16.585 --> 00:08:18.605
S equals one, so that's just one,

00:08:18.605 --> 00:08:21.062
and then over here this
looks like one squared

00:08:21.062 --> 00:08:23.948
minus two times one times one,

00:08:23.948 --> 00:08:25.655
so this will be one minus two.

00:08:25.655 --> 00:08:28.407
That's negative one, so
it is in fact that kind of

00:08:28.407 --> 00:08:30.611
positive, positive, negative
pattern that you were seeing.

00:08:30.611 --> 00:08:32.035
And maybe even from this curve,

00:08:32.035 --> 00:08:34.185
you can get a feel for why the
movement in the X direction

00:08:34.185 --> 00:08:36.878
is twice as much in the movement in the Y.

00:08:36.878 --> 00:08:38.258
It's moving more to the right

00:08:38.258 --> 00:08:40.552
than it is up in the Y direction.

00:08:40.552 --> 00:08:42.370
Then, again, in principle
you can imagine doing this

00:08:42.370 --> 00:08:45.198
not just at the point one,
one but at any given point,

00:08:45.198 --> 00:08:46.633
maybe any given point along this curve

00:08:46.633 --> 00:08:48.759
or any given point along the surface.

00:08:48.759 --> 00:08:50.930
And the corresponding movement,

00:08:50.930 --> 00:08:53.445
the direction that nudges
in the T direction,

00:08:53.445 --> 00:08:56.127
will give you some vector
in three-dimensional space,

00:08:56.127 --> 00:08:57.265
and that's the interpretation.

00:08:57.265 --> 00:09:01.456
That is the meaning of
the partial derivative

00:09:01.456 --> 00:09:04.373
of the vector-valued function here.

00:09:05.750 --> 00:09:09.562
And, again, it's not the
actual nudge vector itself,

00:09:09.562 --> 00:09:11.022
when you nudge the input

00:09:11.022 --> 00:09:14.093
and you get just a little
smidgen in the output space here,

00:09:14.093 --> 00:09:16.756
but it's that divided by
the size of the nudge,

00:09:16.756 --> 00:09:18.493
so that's why you'll get
kind of normal-sized vectors

00:09:18.493 --> 00:09:20.434
rather than tiny vectors.

00:09:20.434 --> 00:09:22.205
And in the next video, I'll
do kind of the same thing

00:09:22.205 --> 00:09:24.708
for what happens when you
nudge in the S direction

00:09:24.708 --> 00:09:26.782
just to get a better
feel for what's going on

00:09:26.782 --> 00:09:28.115
in this example.


Partial derivatives of vector fields

WEBVTT
Kind: captions
Language: en

00:00:01.515 --> 00:00:03.001
- [Voiceover] So let's
start thinking about partial

00:00:03.001 --> 00:00:05.129
derivatives of vector fields.

00:00:05.129 --> 00:00:06.927
So a vector field is a function.

00:00:06.927 --> 00:00:09.626
I'll just do a two
dimensional example here.

00:00:09.626 --> 00:00:12.208
It's gonna be something that
has a two dimensional input.

00:00:12.208 --> 00:00:14.691
And then the output has the
same number of dimensions.

00:00:14.691 --> 00:00:16.048
That's the important part.

00:00:16.048 --> 00:00:17.801
And each of these components in the output

00:00:17.801 --> 00:00:21.016
is gonna depend somehow
on the input variables.

00:00:21.016 --> 00:00:22.791
So the example I have in mind

00:00:22.791 --> 00:00:25.273
will be X times Y as that first component,

00:00:25.273 --> 00:00:29.440
and then Y squared minus X
squared as that second component.

00:00:31.171 --> 00:00:32.985
And you can compute the partial derivative

00:00:32.985 --> 00:00:34.513
of a guy like this, right?

00:00:34.513 --> 00:00:36.417
You'll take the partial
derivative with respect to

00:00:36.417 --> 00:00:37.281
one of the input variables.

00:00:37.281 --> 00:00:38.313
I'll choose X.

00:00:38.313 --> 00:00:39.739
It's always a nice one to start with.

00:00:39.739 --> 00:00:42.822
Partial derivative with respect to X.

00:00:44.235 --> 00:00:45.337
And if we were to actually compute it,

00:00:45.337 --> 00:00:48.607
in this case, it's another,
it's a function of X and Y.

00:00:48.607 --> 00:00:51.106
What you do is you take
the partial derivative

00:00:51.106 --> 00:00:52.363
component wise.

00:00:52.363 --> 00:00:54.095
So you to each component in the first one.

00:00:54.095 --> 00:00:55.016
You say, okay.

00:00:55.016 --> 00:00:55.978
X looks like a variable.

00:00:55.978 --> 00:00:57.635
Y looks like a constant.

00:00:57.635 --> 00:00:59.746
The derivative will just be that constant.

00:00:59.746 --> 00:01:01.984
And then the partial derivative
of the second component.

00:01:01.984 --> 00:01:03.663
That Y squared looks like a constant.

00:01:03.663 --> 00:01:06.266
Derivative of negative X
squared with respect to X.

00:01:06.266 --> 00:01:07.516
Negative two X.

00:01:08.649 --> 00:01:09.994
So analytically,

00:01:09.994 --> 00:01:11.435
if you know how to take
a partial derivative,

00:01:11.435 --> 00:01:12.865
you already know how to
take a partial derivative

00:01:12.865 --> 00:01:15.472
of vector valued functions
and hence vector fields,

00:01:15.472 --> 00:01:17.931
but the fun part, the important part here.

00:01:17.931 --> 00:01:20.354
How do you actually interpret this?

00:01:20.354 --> 00:01:21.468
And this has everything to do

00:01:21.468 --> 00:01:23.370
with visualizing it in some way.

00:01:23.370 --> 00:01:26.762
So the vector field, the reason
we call it a vector field,

00:01:26.762 --> 00:01:29.894
is you kind of take the whole X Y plane

00:01:29.894 --> 00:01:32.007
and you're gonna fill it with vectors.

00:01:32.007 --> 00:01:33.938
And concretely, what I mean by that,

00:01:33.938 --> 00:01:35.022
you'll take a given input.

00:01:35.022 --> 00:01:37.954
What's an input you wanna look at?

00:01:37.954 --> 00:01:39.927
I'll say, maybe one, two.

00:01:39.927 --> 00:01:41.016
Yeah, let's do that.

00:01:41.016 --> 00:01:42.516
Let's do one, two.

00:01:43.473 --> 00:01:45.870
Which would mean you
kinda go X equals one.

00:01:45.870 --> 00:01:47.399
And then Y equals two

00:01:47.399 --> 00:01:48.429
This input point.

00:01:48.429 --> 00:01:49.726
Then we want to associate that

00:01:49.726 --> 00:01:51.781
with the output vector in some way.

00:01:51.781 --> 00:01:54.477
And so, let's just compute
what it should equal.

00:01:54.477 --> 00:01:57.532
So when we plug in X equals
one, and Y equals two,

00:01:57.532 --> 00:01:59.365
X times Y becomes two.

00:02:00.490 --> 00:02:02.279
Y squared minus X squared

00:02:02.279 --> 00:02:03.734
becomes two squared minus one squared,

00:02:03.734 --> 00:02:05.988
so four minus one is three.

00:02:05.988 --> 00:02:08.842
So we have this vector two,
three that we want to associate

00:02:08.842 --> 00:02:10.212
with that input point.

00:02:10.212 --> 00:02:11.965
And vector fields, you
just attach the two points.

00:02:11.965 --> 00:02:14.139
I'm gonna take the vector two, three

00:02:14.139 --> 00:02:15.522
and attach it to this guy.

00:02:15.522 --> 00:02:18.591
So we should have an X component of two

00:02:18.591 --> 00:02:21.258
and then a Y component of three.

00:02:22.540 --> 00:02:24.318
So it's going to end up
looking something like,

00:02:24.318 --> 00:02:26.563
let's see, so Y component of three,

00:02:26.563 --> 00:02:28.230
something like this.

00:02:29.634 --> 00:02:31.595
So that'll be the vector and
we attach it to that point.

00:02:31.595 --> 00:02:34.021
And in principal, you do this
to all the different points.

00:02:34.021 --> 00:02:37.015
And if you did, what you'd get
would be something like this.

00:02:37.015 --> 00:02:38.212
And remember when we represent these,

00:02:38.212 --> 00:02:40.773
especially with computers,
it tends to lie,

00:02:40.773 --> 00:02:43.475
where each represented
vector is much, much shorter

00:02:43.475 --> 00:02:45.247
than it should be in reality,

00:02:45.247 --> 00:02:47.624
but you want to squish
them all onto the same page

00:02:47.624 --> 00:02:49.237
so they don't over run each other.

00:02:49.237 --> 00:02:51.421
And here, color is
supposed to give a general,

00:02:51.421 --> 00:02:53.419
vague sense of relative length.

00:02:53.419 --> 00:02:55.462
So ones that are blue should
thought of as much shorter

00:02:55.462 --> 00:02:56.868
than the ones that are yellow,

00:02:56.868 --> 00:02:58.598
but that doesn't really
give a specific thought

00:02:58.598 --> 00:03:01.207
for how long they should be.

00:03:01.207 --> 00:03:02.355
But, for partial derivatives,

00:03:02.355 --> 00:03:04.766
we actually care a lot
about the specifics.

00:03:04.766 --> 00:03:06.685
And if you think back to

00:03:06.685 --> 00:03:08.515
how we interpret partial derivatives

00:03:08.515 --> 00:03:10.022
in a lot of other contexts,

00:03:10.022 --> 00:03:13.030
what want to do is imagine
this partial X here

00:03:13.030 --> 00:03:15.699
as a slight nudge in
the X direction, right.

00:03:15.699 --> 00:03:17.427
So this was our original input,

00:03:17.427 --> 00:03:19.876
so you might imagine just
nudging it a little bit,

00:03:19.876 --> 00:03:22.509
and the size of that nudge, as a number,

00:03:22.509 --> 00:03:24.460
would be your partial X.

00:03:24.460 --> 00:03:26.299
So then the question is,

00:03:26.299 --> 00:03:28.774
what's the resulting change to the output?

00:03:28.774 --> 00:03:30.103
And because the output is a vector,

00:03:30.103 --> 00:03:32.754
the change to the output is
also going to be a vector.

00:03:32.754 --> 00:03:34.607
So what we want is to say

00:03:34.607 --> 00:03:35.970
there's going to be some other vector

00:03:35.970 --> 00:03:37.592
attached to this point, right?

00:03:37.592 --> 00:03:38.560
It's going to look very similar.

00:03:38.560 --> 00:03:39.946
Maybe it looks like,

00:03:39.946 --> 00:03:43.408
maybe it looks something like this.

00:03:43.408 --> 00:03:45.731
So something similar, but
maybe a little bit different.

00:03:45.731 --> 00:03:48.616
And you want to take that
difference in vector form.

00:03:48.616 --> 00:03:50.998
And I'll describe what I mean
by that in just a moment,

00:03:50.998 --> 00:03:53.309
and then divide by the size
of that original nudge.

00:03:53.309 --> 00:03:56.763
So to be much more specific
about what I mean here,

00:03:56.763 --> 00:03:58.292
if you're comparing two different vectors

00:03:58.292 --> 00:04:00.207
and they're rooted in two different spots,

00:04:00.207 --> 00:04:01.255
I think a good way to start

00:04:01.255 --> 00:04:03.023
is to just move them to a new space

00:04:03.023 --> 00:04:04.803
where they're rooted in the same spot.

00:04:04.803 --> 00:04:05.834
So in this case,

00:04:05.834 --> 00:04:08.513
I'm gonna kinda just draw
a separate space over here.

00:04:08.513 --> 00:04:10.274
And be thinking of this as a place

00:04:10.274 --> 00:04:11.555
for these vectors to live.

00:04:11.555 --> 00:04:13.574
And I'm gonna them both on this plane,

00:04:13.574 --> 00:04:15.600
but I'm gonna root them
each in the origin.

00:04:15.600 --> 00:04:17.990
So this first one that
has components two, three,

00:04:17.990 --> 00:04:19.457
now let's give it a name, right.

00:04:19.457 --> 00:04:21.638
Let's call this guy V one, okay.

00:04:21.638 --> 00:04:22.878
So that'll be V one.

00:04:22.878 --> 00:04:24.912
And then the nudged
output, the second one,

00:04:24.912 --> 00:04:26.245
I'll call V two.

00:04:31.014 --> 00:04:32.784
And let's say V two is also in this space,

00:04:32.784 --> 00:04:34.526
and I might exaggerate the difference,

00:04:34.526 --> 00:04:36.784
just so we can see it here.

00:04:36.784 --> 00:04:38.905
Let's say it was different in some way.

00:04:38.905 --> 00:04:40.367
In reality, if it's a small nudge,

00:04:40.367 --> 00:04:42.386
it'll be different in
only a very small way.

00:04:42.386 --> 00:04:45.014
But let's say these were our two vectors.

00:04:45.014 --> 00:04:47.694
The difference between these guys

00:04:47.694 --> 00:04:51.066
is going to be a vector
that connects the tips.

00:04:51.066 --> 00:04:54.066
And I'm gonna call that guy Partial V.

00:04:54.066 --> 00:04:56.240
And the way you can be thinking about this

00:04:56.240 --> 00:05:00.439
is to say that V one, V
one, that original guy,

00:05:00.439 --> 00:05:04.735
plus, that tiny nudge, the
difference between them

00:05:04.735 --> 00:05:05.652
is equal to

00:05:06.951 --> 00:05:08.217
the two,

00:05:08.217 --> 00:05:10.032
you know, the nudged output.

00:05:10.032 --> 00:05:11.197
And in terms of tip to tail with vectors,

00:05:11.197 --> 00:05:12.261
you've seen that.

00:05:12.261 --> 00:05:14.719
Kind of the green vector
plus that blue vector

00:05:14.719 --> 00:05:17.165
is the same as that pink vector

00:05:17.165 --> 00:05:19.417
that connects the tail of the original one

00:05:19.417 --> 00:05:21.263
to the tip of the new one.

00:05:21.263 --> 00:05:23.824
So when we're thinking
of a partial derivative,

00:05:23.824 --> 00:05:24.741
you're basically saying,

00:05:24.741 --> 00:05:27.301
"Hey, what happens if
we take this, the nudge,

00:05:27.301 --> 00:05:29.302
the size of the nudge of the output,

00:05:29.302 --> 00:05:32.343
and then we divide it by
the nudge of the input?"

00:05:32.343 --> 00:05:34.415
So let's thinking of that
original nudge as being,

00:05:34.415 --> 00:05:38.456
I don't know, of a size of one
half, like zero point five.

00:05:38.456 --> 00:05:41.631
As the change in the X direction.

00:05:41.631 --> 00:05:43.637
Then that would mean, when
you go over here and say

00:05:43.637 --> 00:05:44.644
what's that DV?

00:05:44.644 --> 00:05:48.121
That changing vector V divided by the X,

00:05:48.121 --> 00:05:50.249
you'd be dividing it by zero point five,

00:05:50.249 --> 00:05:51.993
and in principle, you'd be thinking of,

00:05:51.993 --> 00:05:54.603
that would mean that your
kind of scaling this by two.

00:05:54.603 --> 00:05:58.399
As if to say, this
little DV is one half of

00:05:58.399 --> 00:05:59.383
some other vector.

00:05:59.383 --> 00:06:01.951
And that other vector is what
the partial derivative is.

00:06:01.951 --> 00:06:04.263
So this other vector
here, the full blue guy,

00:06:04.263 --> 00:06:08.233
would be DV, you know,
scaled down or scaled up,

00:06:08.233 --> 00:06:09.387
however you want to think about it,

00:06:09.387 --> 00:06:10.703
by that partial X.

00:06:10.703 --> 00:06:12.015
And that's what makes it such the,

00:06:12.015 --> 00:06:13.818
you know, in principle if this partial X

00:06:13.818 --> 00:06:16.055
change was really small, like one 100th

00:06:16.055 --> 00:06:17.830
and the output nudge
was also really small,

00:06:17.830 --> 00:06:19.087
it's like one 100th,

00:06:19.087 --> 00:06:20.676
or you know something on that order,

00:06:20.676 --> 00:06:22.508
it wouldn't be specifically that.

00:06:22.508 --> 00:06:24.227
Then the DVDX, that change,

00:06:24.227 --> 00:06:27.003
would still be a normal sized vector.

00:06:27.003 --> 00:06:28.389
And the direction that it points

00:06:28.389 --> 00:06:29.731
is still kind of an indication

00:06:29.731 --> 00:06:31.782
of the direction that this
green vector should change

00:06:31.782 --> 00:06:34.384
as you're scooting over.

00:06:34.384 --> 00:06:37.355
So just be concrete and
actually compute this guy,

00:06:37.355 --> 00:06:39.639
let's say we were to take
this partial derivative

00:06:39.639 --> 00:06:41.542
partial V with respect to X

00:06:41.542 --> 00:06:43.931
and evaluate it at that point one, two

00:06:43.931 --> 00:06:46.739
that we're just dealing with, one, two.

00:06:46.739 --> 00:06:49.215
What that would mean, Y is equal to two,

00:06:49.215 --> 00:06:51.052
so that first component is two

00:06:51.052 --> 00:06:52.366
and then X is equal to one,

00:06:52.366 --> 00:06:54.255
so that next should be negative two.

00:06:54.255 --> 00:06:57.621
And then we can see just
how wrong my drawing was

00:06:57.621 --> 00:06:58.454
to start here.

00:06:58.454 --> 00:07:00.532
I was just kind of guessing
what the pink vector would be,

00:07:00.532 --> 00:07:01.365
but I guess it changes

00:07:01.365 --> 00:07:03.118
in the direction of two, negative two.

00:07:03.118 --> 00:07:06.762
So that should be something,
here I'll erase the,

00:07:06.762 --> 00:07:10.679
what turns out to be,
the wrong direction here.

00:07:11.743 --> 00:07:13.410
Get rid of this guy.

00:07:15.606 --> 00:07:18.564
And I guess the change would
be in the direction kind of two

00:07:18.564 --> 00:07:20.052
as the X component,

00:07:20.052 --> 00:07:22.883
and then negative two,
that's a negative two,

00:07:22.883 --> 00:07:24.126
as the Y component.

00:07:24.126 --> 00:07:27.211
So the derivative vector should
look something like this.

00:07:27.211 --> 00:07:29.772
Which means all corresponding
little DV nudges,

00:07:29.772 --> 00:07:32.373
will be slight changes,

00:07:32.373 --> 00:07:33.787
will be slight changes on that.

00:07:33.787 --> 00:07:35.757
So these will be your, your DVs.

00:07:35.757 --> 00:07:38.078
Something in that direction.

00:07:38.078 --> 00:07:41.828
And what that means in
our vector field then,

00:07:43.958 --> 00:07:45.752
as you move in the X direction

00:07:45.752 --> 00:07:48.758
and consider the various
vectors attached to each point,

00:07:48.758 --> 00:07:51.411
as you kind of passing
through the point one, two,

00:07:51.411 --> 00:07:53.222
the way that the vectors are changing,

00:07:53.222 --> 00:07:55.373
should be somehow down and to the right.

00:07:55.373 --> 00:07:57.030
The tip should move down and to the right.

00:07:57.030 --> 00:07:59.413
So if this starts highly
up and to the right,

00:07:59.413 --> 00:08:01.184
then it should be getting kind of shorter,

00:08:01.184 --> 00:08:02.706
but then longer to the right.

00:08:02.706 --> 00:08:03.853
So then the V two,

00:08:03.853 --> 00:08:06.382
if I were to have drawn
it more accurately here,

00:08:06.382 --> 00:08:08.824
you know, what they nudged
output should look like,

00:08:08.824 --> 00:08:11.602
it would really be
something that kind of like

00:08:11.602 --> 00:08:12.535
I don't know, like this,

00:08:12.535 --> 00:08:15.693
where it's getting shorter
in the Y direction,

00:08:15.693 --> 00:08:17.829
but then longer in X direction,

00:08:17.829 --> 00:08:20.886
as per that blue nudging arrow.

00:08:20.886 --> 00:08:23.334
And then in the next video I'll kind of go

00:08:23.334 --> 00:08:25.286
through more examples of
how you might think of this.

00:08:25.286 --> 00:08:27.913
How you think of it in terms
of what each component means.

00:08:27.913 --> 00:08:30.489
Which becomes very
important for later topics,

00:08:30.489 --> 00:08:32.278
like divergence and curl.

00:08:32.278 --> 00:08:33.643
And I'll see you next video.


Partial derivatives of vector fields, component by component

WEBVTT
Kind: captions
Language: en

00:00:00.507 --> 00:00:01.653
- [Voiceover] Let's
continue thinking about

00:00:01.653 --> 00:00:04.212
partial derivatives of vector fields.

00:00:04.212 --> 00:00:06.101
This is one of those things
that's pretty good practice

00:00:06.101 --> 00:00:07.512
for some important concepts coming up

00:00:07.512 --> 00:00:09.574
in multivariable calc,
and it's also just good

00:00:09.574 --> 00:00:11.759
to sit down and take a complicated thing

00:00:11.759 --> 00:00:14.344
and kind of break it down piece by piece.

00:00:14.344 --> 00:00:17.030
So a vector field like
the one I just showed

00:00:17.030 --> 00:00:19.586
is represented by a
vector-valued function,

00:00:19.586 --> 00:00:21.696
and since it's two-dimensional, it'll have

00:00:21.696 --> 00:00:23.905
some kind of two-dimensional input.

00:00:23.905 --> 00:00:26.081
And the output will be a vector,

00:00:26.081 --> 00:00:28.725
each of whose components
is some kind of function

00:00:28.725 --> 00:00:29.872
of x and y, right?

00:00:29.872 --> 00:00:33.962
So I'll just write P of
x, y for that x-component

00:00:33.962 --> 00:00:36.879
and Q of x, y for that y-component.

00:00:38.045 --> 00:00:42.437
And each of these are just
scalar-valued functions.

00:00:42.437 --> 00:00:45.513
It's actually quite common to
use P and Q for these values.

00:00:45.513 --> 00:00:47.749
It's one of those things where
sometimes you'll even see

00:00:47.749 --> 00:00:49.378
a theorem about vector calculus

00:00:49.378 --> 00:00:50.615
in terms of just P and Q,

00:00:50.615 --> 00:00:52.658
kind of leaving it understood
to the reader that,

00:00:52.658 --> 00:00:55.398
yeah, P and Q always refer
to the x and y components

00:00:55.398 --> 00:00:57.448
of the output of a vector field.

00:00:57.448 --> 00:00:59.985
And in this specific case,
the function that I chose,

00:00:59.985 --> 00:01:01.842
it's actually the one that
I used in the last video.

00:01:01.842 --> 00:01:04.486
P is equal to x times y,

00:01:04.486 --> 00:01:08.403
and Q is equal to y
squared minus x squared.

00:01:10.245 --> 00:01:11.111
And in the last video,

00:01:11.111 --> 00:01:13.285
I was talking about interpreting
the partial derivative

00:01:13.285 --> 00:01:15.045
of v, the vector-valued function

00:01:15.045 --> 00:01:17.202
with respect to one of the variables,

00:01:17.202 --> 00:01:18.734
which has its merits, and
I think it's a good way

00:01:18.734 --> 00:01:21.810
to understand vector-valued
functions in general.

00:01:21.810 --> 00:01:23.800
But here that's not what I'm gonna do.

00:01:23.800 --> 00:01:25.871
It's actually, another useful skill

00:01:25.871 --> 00:01:28.852
is to just think in terms
of each specific component.

00:01:28.852 --> 00:01:30.249
So if we just think of P and Q,

00:01:30.249 --> 00:01:33.166
we have four possible partial derivatives

00:01:33.166 --> 00:01:35.918
at our disposal here, two
of them with respect to P.

00:01:35.918 --> 00:01:38.053
So you can think about
the partial derivative

00:01:38.053 --> 00:01:40.199
of P with respect to x,

00:01:40.199 --> 00:01:43.809
or the partial derivative
of P with respect to y.

00:01:43.809 --> 00:01:45.040
And then similarly, Q,

00:01:45.040 --> 00:01:47.508
you could think about
partial derivative of Q

00:01:47.508 --> 00:01:49.008
with respect to x.

00:01:50.621 --> 00:01:52.793
This should be a partial.

00:01:52.793 --> 00:01:55.029
Or the partial derivative of Q

00:01:55.029 --> 00:01:57.295
with respect to y.

00:01:57.295 --> 00:01:59.330
So four different values
that you could be looking at

00:01:59.330 --> 00:02:01.613
and considering and
understanding how they influence

00:02:01.613 --> 00:02:03.857
the change of the vector field as a whole.

00:02:03.857 --> 00:02:05.075
And in this specific example,

00:02:05.075 --> 00:02:06.377
let's actually compute these.

00:02:06.377 --> 00:02:09.175
So derivative of P with respect to x.

00:02:09.175 --> 00:02:10.751
P is this first component.

00:02:10.751 --> 00:02:13.102
We're taking the partial
of this with respect to x.

00:02:13.102 --> 00:02:14.426
y looks like a constant.

00:02:14.426 --> 00:02:15.843
Constant times x.

00:02:17.036 --> 00:02:18.594
Derivative is just that constant.

00:02:18.594 --> 00:02:20.552
If we took the derivative
with respect to y,

00:02:20.552 --> 00:02:21.575
the roles have reversed,

00:02:21.575 --> 00:02:23.148
and its partial derivative is x,

00:02:23.148 --> 00:02:25.384
'cause x looks like that constant.

00:02:25.384 --> 00:02:26.989
But Q, its partial derivative

00:02:26.989 --> 00:02:30.045
with respect to x, y
looks like a constant,

00:02:30.045 --> 00:02:33.566
negative x squared goes to negative 2x.

00:02:33.566 --> 00:02:35.551
But then when you're taking
it with respect to y,

00:02:35.551 --> 00:02:38.346
y squared now looks like a function

00:02:38.346 --> 00:02:39.819
whose derivative is 2y,

00:02:39.819 --> 00:02:41.640
and negative x squared
looks like the constant.

00:02:41.640 --> 00:02:44.771
So these are the four
possible partial derivatives.

00:02:44.771 --> 00:02:46.862
But let's actually see
if we can understand

00:02:46.862 --> 00:02:48.800
how they influence the
function as a whole.

00:02:48.800 --> 00:02:50.411
What it means in terms of the picture

00:02:50.411 --> 00:02:51.790
that we're looking at up here.

00:02:51.790 --> 00:02:53.961
And, in particular,
let's focus on a point,

00:02:53.961 --> 00:02:55.501
a specific point,

00:02:55.501 --> 00:02:57.751
and let's do this one here.

00:02:59.886 --> 00:03:01.836
So it's something that's
sitting on the x-axis.

00:03:01.836 --> 00:03:03.533
So this is where y equals zero

00:03:03.533 --> 00:03:05.346
and x is something positive.

00:03:05.346 --> 00:03:09.734
So this is probably when x
is around two-ish, let's say.

00:03:09.734 --> 00:03:11.058
So the value we wanna look at

00:03:11.058 --> 00:03:11.891
is x, y

00:03:14.319 --> 00:03:15.996
when x is two

00:03:15.996 --> 00:03:17.163
and y is zero.

00:03:19.565 --> 00:03:20.815
So if we start plugging that in here,

00:03:20.815 --> 00:03:24.205
what that would mean,
this guy goes to zero,

00:03:24.205 --> 00:03:26.232
this guy goes to two,

00:03:26.232 --> 00:03:28.311
this guy, negative two times x,

00:03:28.311 --> 00:03:29.972
is gonna be negative two.

00:03:29.972 --> 00:03:33.163
And then negative two
times y is gonna be zero.

00:03:33.163 --> 00:03:35.889
And let's start by just looking
at the partial derivative

00:03:35.889 --> 00:03:38.057
of P with respect to x.

00:03:38.057 --> 00:03:40.557
So what that means is
that we're looking for

00:03:40.557 --> 00:03:42.974
how the x-component of
these vectors change

00:03:42.974 --> 00:03:46.463
as you move in the x direction.

00:03:46.463 --> 00:03:48.704
For example, around this
point, we're kind of thinking

00:03:48.704 --> 00:03:50.794
of moving in the x direction, vaguely.

00:03:50.794 --> 00:03:53.118
So we wanna look at the
two neighboring vectors

00:03:53.118 --> 00:03:56.882
and consider what's going
on with the x direction.

00:03:56.882 --> 00:03:59.189
But these vectors, this
one points purely down.

00:03:59.189 --> 00:04:00.443
This one also points purely down,

00:04:00.443 --> 00:04:01.885
and so does this one.

00:04:01.885 --> 00:04:03.327
So no change is happening

00:04:03.327 --> 00:04:05.517
when it comes to the
x-component of these vectors,

00:04:05.517 --> 00:04:09.616
which makes sense because the
value at that point is zero.

00:04:09.616 --> 00:04:11.775
The partial derivative of P
with respect to x is zero,

00:04:11.775 --> 00:04:12.933
so we wouldn't expect a change.

00:04:12.933 --> 00:04:14.766
But on the other hand,

00:04:15.905 --> 00:04:17.580
on the other hand, if we're
looking at partial derivative

00:04:17.580 --> 00:04:19.497
of P with respect to y,

00:04:20.344 --> 00:04:21.696
this should be positive.

00:04:21.696 --> 00:04:25.134
So this should suggest that
the change in the x-component

00:04:25.134 --> 00:04:28.717
as you move in the y
direction is positive.

00:04:29.571 --> 00:04:31.542
So we go up here and now we're not looking

00:04:31.542 --> 00:04:34.698
at change in the x-direction.

00:04:34.698 --> 00:04:38.588
Not looking at change in the x-direction.

00:04:38.588 --> 00:04:39.698
But instead we're wondering what happens

00:04:39.698 --> 00:04:41.752
as we move generally upwards.

00:04:41.752 --> 00:04:44.110
So we're gonna kind of
compare it to these two guys.

00:04:44.110 --> 00:04:46.473
And in that case, the
x-component of this one

00:04:46.473 --> 00:04:48.165
is a little bit to the left.

00:04:48.165 --> 00:04:49.585
The one below it, it's a
little bit to the left.

00:04:49.585 --> 00:04:52.059
Then we get to our main
guy here and it's zero.

00:04:52.059 --> 00:04:53.865
The x-component is zero 'cause
it's pointing purely down.

00:04:53.865 --> 00:04:56.019
And up here it's pointing
a little bit to the right.

00:04:56.019 --> 00:04:57.519
So as y increases,

00:04:58.381 --> 00:05:01.695
the x-component of these
vectors also increases.

00:05:01.695 --> 00:05:03.526
And again, that makes sense

00:05:03.526 --> 00:05:06.366
because this partial
derivative is positive.

00:05:06.366 --> 00:05:08.725
This two suggests that
as you're changing y,

00:05:08.725 --> 00:05:12.362
the value of p, the
x-component of our function,

00:05:12.362 --> 00:05:13.944
should probably keep that on screen.

00:05:13.944 --> 00:05:16.147
You know, the x-component of
our vector-valued function

00:05:16.147 --> 00:05:18.802
is increasing 'cause that's positive.

00:05:18.802 --> 00:05:20.201
For contrast, let's say we look

00:05:20.201 --> 00:05:22.124
at the Q component over here.

00:05:22.124 --> 00:05:23.268
So what this is doing,

00:05:23.268 --> 00:05:24.454
we're looking at changes in the x,

00:05:24.454 --> 00:05:28.317
and we're wondering what the
y value of the vector does.

00:05:28.317 --> 00:05:30.190
So we go up here.

00:05:30.190 --> 00:05:33.584
And now, we're not looking at
changes in the y direction.

00:05:33.584 --> 00:05:35.498
But instead, we're going
back to considering

00:05:35.498 --> 00:05:37.436
what happens as we change x,

00:05:37.436 --> 00:05:40.300
as we're kind of moving in
the horizontal direction here.

00:05:40.300 --> 00:05:42.481
So again, we look at
these neighboring guys.

00:05:42.481 --> 00:05:46.227
And now, the y-component
starts off small, but negative,

00:05:46.227 --> 00:05:47.523
then gets a little bit more negative,

00:05:47.523 --> 00:05:49.141
then gets even more negative.

00:05:49.141 --> 00:05:50.708
And if we kind of keep looking at these,

00:05:50.708 --> 00:05:53.050
the y-component is getting more,
and more, and more negative

00:05:53.050 --> 00:05:55.166
so it's decreasing.

00:05:55.166 --> 00:05:57.564
The value of Q, the
y-component of these vectors

00:05:57.564 --> 00:05:59.290
should be decreasing.

00:05:59.290 --> 00:06:01.472
And that lines up 'cause
the partial derivative here

00:06:01.472 --> 00:06:03.248
was negative two, so that's telling us,

00:06:03.248 --> 00:06:05.927
given that it started negative,
it's getting more negative.

00:06:05.927 --> 00:06:07.288
If it started positive, they would've been

00:06:07.288 --> 00:06:09.212
kind of getting shorter as vectors

00:06:09.212 --> 00:06:11.473
as their y-component got smaller.

00:06:11.473 --> 00:06:13.679
And then finally, just to close things off

00:06:13.679 --> 00:06:16.560
nice and simply here, if
we look at partial of Q

00:06:16.560 --> 00:06:18.060
with respect to y,

00:06:21.021 --> 00:06:23.865
so now if we start looking at
changes in the y direction,

00:06:23.865 --> 00:06:26.588
and we start considering
how, as you move below,

00:06:26.588 --> 00:06:28.081
and then starting to go up,

00:06:28.081 --> 00:06:29.491
what happens to the y-component,

00:06:29.491 --> 00:06:32.348
here, the y-component is a
little bit negative, right?

00:06:32.348 --> 00:06:33.607
It's pointing down and to the left.

00:06:33.607 --> 00:06:35.368
So down, it's a little bit negative.

00:06:35.368 --> 00:06:37.982
Here, the y-component is
also a little bit negative.

00:06:37.982 --> 00:06:40.433
Over here, well, it remains
a little bit negative.

00:06:40.433 --> 00:06:42.487
And you know, from our heuristic look,

00:06:42.487 --> 00:06:44.274
there's no discernible change.

00:06:44.274 --> 00:06:45.314
Maybe it's changing a little bit,

00:06:45.314 --> 00:06:47.343
and we don't have a fine
enough vision of these vectors

00:06:47.343 --> 00:06:49.441
to see that, but if we actually go back

00:06:49.441 --> 00:06:52.843
to the analysis, and see what we computed,

00:06:52.843 --> 00:06:54.027
in fact, it is zero.

00:06:54.027 --> 00:06:55.514
That fact that it looked like

00:06:55.514 --> 00:06:56.675
there was not too much change

00:06:56.675 --> 00:06:59.141
in the y-component of
each one of these vectors

00:06:59.141 --> 00:07:01.413
corresponds with the fact
that the partial derivative

00:07:01.413 --> 00:07:04.107
of that y-component with respect to y,

00:07:04.107 --> 00:07:07.774
with respect to vertical
movements, is zero.

00:07:09.345 --> 00:07:11.290
So this kind of analysis
should give a better feel

00:07:11.290 --> 00:07:13.943
for how we understand the
four different possible

00:07:13.943 --> 00:07:15.859
partial derivatives and what they indicate

00:07:15.859 --> 00:07:18.159
about the vector field,
and you'll get plenty

00:07:18.159 --> 00:07:20.228
of chance to practice that understanding

00:07:20.228 --> 00:07:22.337
as we learn about divergence and curl,

00:07:22.337 --> 00:07:24.617
and try to understand
why each one of those

00:07:24.617 --> 00:07:26.415
represents the thing
that it's supposed to.

00:07:26.415 --> 00:07:30.440
And you'll see what I mean by
that in just a couple videos.


The Gilded Age part 2

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:02.354
- [Voiceover] So, we were talking about

00:00:02.354 --> 00:00:05.502
the wealth inequality that
characterized the Gilded Age,

00:00:05.502 --> 00:00:08.484
but you were telling me that
that's not the only thing,

00:00:08.484 --> 00:00:11.047
Kim, that characterizes this period.

00:00:11.047 --> 00:00:14.781
- [Voiceover] Right, what really
makes the Gilded Age happen

00:00:14.781 --> 00:00:18.864
is what we call the Second
Industrial Revolution.

00:00:20.629 --> 00:00:24.028
Are you familiar with the
First Industrial Revolution?

00:00:24.028 --> 00:00:25.477
- [Voiceover] But, of course.

00:00:25.477 --> 00:00:27.372
- [Voiceover] So, that was the revolution

00:00:27.372 --> 00:00:30.465
where they had steamships and canals

00:00:30.465 --> 00:00:32.749
and kinda this early creation

00:00:32.749 --> 00:00:34.895
of the market system in the United States,

00:00:34.895 --> 00:00:36.645
say like 1820s-1830s.

00:00:37.486 --> 00:00:39.437
The Second Industrial Revolution

00:00:39.437 --> 00:00:43.604
is more of a revolution of
mass production, I would say,

00:00:45.093 --> 00:00:49.176
and ways of making and
shipping and communicating

00:00:52.171 --> 00:00:55.124
about business transactions and materials

00:00:55.124 --> 00:00:56.406
that didn't exist before.

00:00:56.406 --> 00:00:57.994
- [Voiceover] So, what are some of these

00:00:57.994 --> 00:01:02.480
disruptive technologies (Kim
laughs) that are really poised

00:01:02.480 --> 00:01:04.904
to change the shipment paradigm?

00:01:04.904 --> 00:01:08.137
So, okay, so off the top of my head,

00:01:08.137 --> 00:01:11.508
trains probably a huge
deal in this period?

00:01:11.508 --> 00:01:12.341
- [Voiceover] Yes.

00:01:12.341 --> 00:01:14.701
- [Voiceover] So, we've
got all this coal going

00:01:14.701 --> 00:01:17.460
and that means that there's
a lot of smelting happening

00:01:17.460 --> 00:01:18.381
and that means that there's also

00:01:18.381 --> 00:01:20.156
a lot of steel happening too.

00:01:20.156 --> 00:01:21.270
- [Voiceover] Steel.

00:01:21.270 --> 00:01:25.437
I think if I had to choose
one most important technology

00:01:27.372 --> 00:01:30.382
of the Gilded Age, it
would have to be steel.

00:01:30.382 --> 00:01:32.524
Now, it's not like steel didn't
actually exist before this.

00:01:32.524 --> 00:01:35.223
- [Voiceover] Right, steel
had been around for millennia.

00:01:35.223 --> 00:01:36.056
- [Voiceover] Yeah, I think so.

00:01:36.056 --> 00:01:37.504
- [Voiceover] Millennium.

00:01:37.504 --> 00:01:39.399
- [Voiceover] But, what
happens in this time period

00:01:39.399 --> 00:01:41.572
is there's a new process for making steel.

00:01:41.572 --> 00:01:44.637
It's called the Bessemer process.

00:01:44.637 --> 00:01:48.594
And the Bessemer process
basically makes steel faster

00:01:48.594 --> 00:01:51.269
and it makes it cheaper.

00:01:51.269 --> 00:01:54.776
And in this time period,
Andrew Carnegie we talked about

00:01:54.776 --> 00:01:57.005
being this major steel baron,

00:01:57.005 --> 00:01:59.011
railroads throughout the United States,

00:01:59.011 --> 00:02:03.135
partly supported, majorly
supported by the U.S. government.

00:02:03.135 --> 00:02:06.885
And during this period,
they lay 40,000 miles

00:02:08.011 --> 00:02:09.509
of new tracks of rail.

00:02:09.509 --> 00:02:11.067
- [Voiceover] That is so many miles.

00:02:11.067 --> 00:02:12.312
- [Voiceover] That is so many miles.

00:02:12.312 --> 00:02:13.785
- [Voiceover] How long
is the United States

00:02:13.785 --> 00:02:15.286
from Los Angeles to New York? (Kim laughs)

00:02:15.286 --> 00:02:16.875
Like 3,000 miles?

00:02:16.875 --> 00:02:18.348
3,100 miles?

00:02:18.348 --> 00:02:19.181
- [Voiceover] Yes.

00:02:19.181 --> 00:02:22.328
So, just imagine a nation
where most railroad tracks

00:02:22.328 --> 00:02:25.226
had gone through sort of
the eastern coastal cities

00:02:25.226 --> 00:02:26.954
up until 1865.

00:02:26.954 --> 00:02:29.824
Now, the entire country
is connected by rail.

00:02:29.824 --> 00:02:34.226
There's more rail in the
United States in 1900

00:02:34.226 --> 00:02:36.372
than all of Europe combined.

00:02:36.372 --> 00:02:39.604
- [Voiceover] So, the Bessemer
process of making steel

00:02:39.604 --> 00:02:41.527
is this foundational technology

00:02:41.527 --> 00:02:43.756
that enables a lot of
the Gilded Age to happen.

00:02:43.756 --> 00:02:45.734
- [Voiceover] Right, so it
enables the United States

00:02:45.734 --> 00:02:49.189
to move out and also to connect markets.

00:02:49.189 --> 00:02:52.477
So, you can now take raw
materials from the West,

00:02:52.477 --> 00:02:55.319
which is really important,
that's where the gold lives

00:02:55.319 --> 00:02:57.744
and also cattle ranching,

00:02:57.744 --> 00:03:00.753
you take those things from the West,

00:03:00.753 --> 00:03:04.372
you take them to the
cities to be processed.

00:03:04.372 --> 00:03:06.573
Then, you take the finished goods

00:03:06.573 --> 00:03:10.223
and send them back out
into the smaller towns.

00:03:10.223 --> 00:03:12.452
So, rail facilitates all of that.

00:03:12.452 --> 00:03:16.130
- [Voiceover] So, this is how
my hometown became notorious--

00:03:16.130 --> 00:03:16.963
- [Voiceover] Yeah.

00:03:16.963 --> 00:03:17.796
- [Voiceover] of Chicago.

00:03:17.796 --> 00:03:19.080
So, there would be cattle
drives, I guess then,

00:03:19.080 --> 00:03:20.692
that came from the West

00:03:20.692 --> 00:03:22.026
and then they would all be slaughtered

00:03:22.026 --> 00:03:23.644
and processed in Chicago.

00:03:23.644 --> 00:03:25.171
- [Voiceover] Right, yeah.

00:03:25.171 --> 00:03:27.874
I'm a native Pennsylvanian,
you're a native Chicagoan,

00:03:27.874 --> 00:03:30.744
and we are born from steel places,

00:03:30.744 --> 00:03:35.342
as the steel industry really
grew up in Pittsburgh.

00:03:35.342 --> 00:03:37.403
And what I think is really
interesting about steel too

00:03:37.403 --> 00:03:41.026
is that it's like a
self-sustaining industry

00:03:41.026 --> 00:03:45.109
because you need the steel
to make the railroads.

00:03:47.100 --> 00:03:49.683
And then, the railroad industry

00:03:50.918 --> 00:03:54.289
pays for the creation of steel,

00:03:54.289 --> 00:03:58.274
which facilitates the
creation of more railroads,

00:03:58.274 --> 00:04:00.921
which necessitates the
creation of more steel

00:04:00.921 --> 00:04:03.345
and it's just like this never-ending boom

00:04:03.345 --> 00:04:05.072
in steel in the Gilded Age.

00:04:05.072 --> 00:04:09.754
So, steel facilitates the
United States moving outward,

00:04:09.754 --> 00:04:13.041
but it also facilitates the
United States moving upward.

00:04:13.041 --> 00:04:13.874
- [Voiceover] Oh, I see what you did.

00:04:13.874 --> 00:04:14.707
That was good.

00:04:14.707 --> 00:04:15.540
- [Voiceover] Yeah, you like that?

00:04:15.540 --> 00:04:16.373
- [Voiceover] Yeah.

00:04:16.373 --> 00:04:19.240
- [Voiceover] So, steel
allows for the construction

00:04:19.240 --> 00:04:22.084
of buildings that are
taller than ever before.

00:04:22.084 --> 00:04:23.500
- [Voiceover] So, what
is this building here?

00:04:23.500 --> 00:04:24.333
- [Voiceover] This building here

00:04:24.333 --> 00:04:27.059
is the Home Insurance Building in Chicago.

00:04:27.059 --> 00:04:30.375
I don't believe it is there anymore.

00:04:30.375 --> 00:04:31.740
And do you know what is special

00:04:31.740 --> 00:04:33.273
about the Home Insurance Building?

00:04:33.273 --> 00:04:34.638
- [Voiceover] No, what's special about it?

00:04:34.638 --> 00:04:36.028
- [Voiceover] The Home
Insurance Building is considered

00:04:36.028 --> 00:04:37.783
to be the world's first skyscraper.

00:04:37.783 --> 00:04:39.005
- [Voiceover] What?! Awesome!

00:04:39.005 --> 00:04:40.674
- [Voiceover] Yeah, it looks pretty short

00:04:40.674 --> 00:04:42.704
for a skyscraper by modern standards.

00:04:42.704 --> 00:04:44.822
- [Voiceover] I mean, I couldn't
build a building that tall.

00:04:44.822 --> 00:04:46.772
- [Voiceover] That's 10 stories tall.

00:04:46.772 --> 00:04:48.751
And what you can do with steel

00:04:48.751 --> 00:04:52.918
is build these steel-frame
structures that allow you,

00:04:54.543 --> 00:04:58.710
without using stone, there's
kind of like a steel cage

00:05:00.533 --> 00:05:03.097
underneath the facade of this.

00:05:03.097 --> 00:05:05.605
And so, you can build buildings

00:05:05.605 --> 00:05:08.688
that are taller while having windows.

00:05:10.202 --> 00:05:11.345
- [Voiceover] It's like a Faraday cage.

00:05:11.345 --> 00:05:13.211
I bet there was terrible
cellphone reception in there.

00:05:13.211 --> 00:05:14.772
- [Voiceover] I imagine so, yeah.

00:05:14.772 --> 00:05:16.388
There were no cellphones at this time.

00:05:16.388 --> 00:05:18.116
All right, so you know what else

00:05:18.116 --> 00:05:19.592
made these tall buildings possible

00:05:19.592 --> 00:05:21.682
except for the steel structures?

00:05:21.682 --> 00:05:22.515
- [Voiceover] Was it elevators?

00:05:22.515 --> 00:05:23.856
- [Voiceover] It was totally elevators.

00:05:23.856 --> 00:05:24.886
- [Voiceover] Yes!

00:05:24.886 --> 00:05:26.419
- [Voiceover] Yes, see, you're better

00:05:26.419 --> 00:05:27.366
at this than you thought.

00:05:27.366 --> 00:05:28.369
- [Voiceover] Yeah!

00:05:28.369 --> 00:05:29.508
- [Voiceover] This is the time

00:05:29.508 --> 00:05:32.378
of the invention of the Otis elevator,

00:05:32.378 --> 00:05:34.635
and this is my little
elevator entrance, that--

00:05:34.635 --> 00:05:35.471
- [Voiceover] Nice!

00:05:35.471 --> 00:05:37.808
- [Voiceover] You could go
to the top of a tall building

00:05:37.808 --> 00:05:40.817
without having to walk
up 37 flights of stairs,

00:05:40.817 --> 00:05:43.046
which is pretty sweet for our efficiency,

00:05:43.046 --> 00:05:44.690
if not maybe our waistlines.

00:05:44.690 --> 00:05:47.059
- [Voiceover] Okay, so, but
we've got this steel process

00:05:47.059 --> 00:05:50.765
which enable the construction
of tons and tons of rail

00:05:50.765 --> 00:05:54.750
and tons and tons of
buildings, of new buildings

00:05:54.750 --> 00:05:58.141
where you can put more
industry and more people.

00:05:58.141 --> 00:06:01.624
And that enable cities to
grow and wealth to grow.

00:06:01.624 --> 00:06:02.906
- [Voiceover] Yeah, exactly.

00:06:02.906 --> 00:06:05.525
So, there are more and more
people flooding into cities.

00:06:05.525 --> 00:06:08.025
By 1870, there are more people

00:06:09.064 --> 00:06:12.519
working for other people
for wages living in cities

00:06:12.519 --> 00:06:14.470
than people who work for themselves,

00:06:14.470 --> 00:06:18.900
which is a new era in the
American economic system.

00:06:18.900 --> 00:06:22.467
There's some other really
important business technologies

00:06:22.467 --> 00:06:24.166
that grow up in this time period as well.

00:06:24.166 --> 00:06:26.333
So, there's the telephone,

00:06:29.600 --> 00:06:32.498
which makes it possible to do

00:06:32.498 --> 00:06:36.038
business transactions on the spot.

00:06:36.038 --> 00:06:38.453
It revolutionized the speed of business

00:06:38.453 --> 00:06:40.209
very much the same way that the internet

00:06:40.209 --> 00:06:44.026
is gonna revolutionize the
speed of business in the 1990s.

00:06:44.026 --> 00:06:47.648
You also have refrigeration,
which you would not think

00:06:47.648 --> 00:06:49.961
would be that big of a deal,

00:06:49.961 --> 00:06:54.128
but think about how it
allows you to move foodstuffs

00:06:55.255 --> 00:06:56.983
all over the country to new markets.

00:06:56.983 --> 00:06:59.097
You were just talking
about Chicago, right?

00:06:59.097 --> 00:07:02.580
So, the only way that cattle
could be driven into Chicago,

00:07:02.580 --> 00:07:04.809
slaughtered, and then have meat sent

00:07:04.809 --> 00:07:07.010
to all the other markets
in the United States

00:07:07.010 --> 00:07:09.128
was through refrigerated train cars.

00:07:09.128 --> 00:07:10.995
And they have similar
things for steamships

00:07:10.995 --> 00:07:12.444
that allow people to, for example,

00:07:12.444 --> 00:07:14.868
bring oranges from Florida to New York.

00:07:14.868 --> 00:07:17.375
So, it's this web of markets

00:07:17.375 --> 00:07:19.577
that are connecting the United States,

00:07:19.577 --> 00:07:22.244
and this is my terrible
drawing of the United States,

00:07:22.244 --> 00:07:25.577
but rail and then ships make it possible

00:07:28.176 --> 00:07:32.104
for all of these markets
to connect together

00:07:32.104 --> 00:07:34.262
over time and over space.

00:07:34.262 --> 00:07:35.236
- [Voiceover] That's super cool.

00:07:35.236 --> 00:07:37.711
- [Voiceover] You know, the
railroad was even so important

00:07:37.711 --> 00:07:39.856
in this time period that, in a way,

00:07:39.856 --> 00:07:43.312
it invented the modern system of time.

00:07:43.312 --> 00:07:46.098
Because before the railroad,

00:07:46.098 --> 00:07:48.745
localities would just decide when noon was

00:07:48.745 --> 00:07:51.253
based on when the sun
was highest in the sky,

00:07:51.253 --> 00:07:52.086
which meant that--

00:07:52.086 --> 00:07:53.363
- [Voiceover] It didn't
matter whether or not

00:07:53.363 --> 00:07:56.985
it was the same time in
Kansas City as in St. Louis.

00:07:56.985 --> 00:07:59.549
But once you have a train connecting 'em,

00:07:59.549 --> 00:08:02.029
the St. Louis train gets in at 12:05,

00:08:02.029 --> 00:08:03.951
if you're off, you'll miss your train.

00:08:03.951 --> 00:08:07.156
- [Voiceover] Or that might
lead to a collision of trains

00:08:07.156 --> 00:08:08.629
if they don't know when the other train

00:08:08.629 --> 00:08:09.990
is going to be coming through.

00:08:09.990 --> 00:08:11.077
- [Voiceover] Yeah, okay.

00:08:11.077 --> 00:08:13.497
So, inventions of the Gilded Age:

00:08:13.497 --> 00:08:17.426
intranational train travel, the telephone,

00:08:17.426 --> 00:08:20.101
refrigeration for meat, the
Bessemer process for steel,

00:08:20.101 --> 00:08:21.968
and the standardization of time.

00:08:21.968 --> 00:08:23.445
- [Voiceover] All of those things.

00:08:23.445 --> 00:08:24.755
And I would say the last thing

00:08:24.755 --> 00:08:27.931
that might be really important
here is also electrification.

00:08:27.931 --> 00:08:29.352
- [Voiceover] Oooh!

00:08:29.352 --> 00:08:32.939
- [Voiceover] Yeah, and like steel,

00:08:32.939 --> 00:08:35.781
electricity was not
invented in the Gilded Age,

00:08:35.781 --> 00:08:39.567
but what happened was the
spread of the light bulb

00:08:39.567 --> 00:08:41.963
in both homes and businesses,

00:08:41.963 --> 00:08:44.861
which meant that you
could work longer hours.

00:08:44.861 --> 00:08:45.944
You didn't sleep as long.

00:08:45.944 --> 00:08:49.061
Actually, the amount of sleep
that people got per night

00:08:49.061 --> 00:08:52.489
switched from about nine
hours before electrification

00:08:52.489 --> 00:08:54.328
to about seven hours after.

00:08:54.328 --> 00:08:56.557
So, Thomas Edison is literally responsible

00:08:56.557 --> 00:08:58.034
for robbing us of sleep.

00:08:58.034 --> 00:08:59.674
But it also made it possible

00:08:59.674 --> 00:09:02.014
for workers to work longer hours

00:09:02.014 --> 00:09:06.181
and it significantly reduced
the risk of fire in businesses,

00:09:07.727 --> 00:09:10.401
which meant you could invest
in them with more confidence.

00:09:10.401 --> 00:09:12.129
- [Voiceover] Oh, because
they didn't have gas lamps

00:09:12.129 --> 00:09:13.244
that could burst into flame.

00:09:13.244 --> 00:09:14.077
- [Voiceover] Right.

00:09:14.077 --> 00:09:15.967
- [Voiceover] Awesome,
well that's super cool.

00:09:15.967 --> 00:09:17.945
So, hooray Gilded Age, right?

00:09:17.945 --> 00:09:18.948
- [Voiceover] Right.

00:09:18.948 --> 00:09:22.626
And I think one thing that's
important to understand

00:09:22.626 --> 00:09:27.113
about these technologies
is that one of the goals

00:09:27.113 --> 00:09:29.927
of these technologies
was to make it possible

00:09:29.927 --> 00:09:32.017
to produce things faster,

00:09:32.017 --> 00:09:36.885
but also to produce them
with less-skilled workers,

00:09:36.885 --> 00:09:40.870
because a skilled worker,
someone who knows a craft

00:09:40.870 --> 00:09:44.771
and can produce a finished
item from start to finish,

00:09:44.771 --> 00:09:47.554
that takes a long time and
it costs a lot of money.

00:09:47.554 --> 00:09:50.842
This is the difference between
buying a suit off the rack

00:09:50.842 --> 00:09:52.792
and having a bespoke suit.

00:09:52.792 --> 00:09:56.076
If you wanna pay someone
for that time and talent,

00:09:56.076 --> 00:09:57.692
you're gonna pay a lot.

00:09:57.692 --> 00:10:00.590
But if you can make
something on a machine,

00:10:00.590 --> 00:10:03.739
then you can make a lot
of them very quickly

00:10:03.739 --> 00:10:08.444
and you don't need someone
who is an expert tailor.

00:10:08.444 --> 00:10:11.008
You just need someone who
can operate a sewing machine

00:10:11.008 --> 00:10:13.571
to do a couple of seams.

00:10:13.571 --> 00:10:16.971
So, what they're trying to
do here with this innovation

00:10:16.971 --> 00:10:19.646
is spread consumer goods,

00:10:19.646 --> 00:10:21.986
spread a higher standard of living.

00:10:21.986 --> 00:10:24.773
But they're also doing that at the price

00:10:24.773 --> 00:10:28.932
of having less-skilled workers
making much smaller wages.


The Gilded Age part 1

WEBVTT
Kind: captions
Language: en

00:00:00.085 --> 00:00:01.254
- [Voiceover] Hello David.

00:00:01.254 --> 00:00:02.545
- [Voiceover] Hello Kim.

00:00:02.545 --> 00:00:03.378
- [Voiceover] So I brought you here

00:00:03.378 --> 00:00:04.406
to talk about the Gilded Age,

00:00:04.406 --> 00:00:06.634
which is one of my favorite eras

00:00:06.634 --> 00:00:08.016
of American history.

00:00:08.016 --> 00:00:08.933
- [Voiceover] Because everything was great

00:00:08.933 --> 00:00:09.955
and covered in gold?

00:00:09.955 --> 00:00:11.998
- [Voiceover] No, because
it is the only era

00:00:11.998 --> 00:00:13.716
of American history I can think of

00:00:13.716 --> 00:00:15.818
that has a sarcastic name.

00:00:15.818 --> 00:00:16.909
(laughter)

00:00:16.909 --> 00:00:19.011
- [Voiceover] What's
to be sarcastic about?

00:00:19.011 --> 00:00:20.450
What's happening between,

00:00:20.450 --> 00:00:23.155
so I see it's from 1865 to 1898

00:00:23.155 --> 00:00:25.547
which is the end of the Civil War,

00:00:25.547 --> 00:00:27.230
and then what happens in 1898?

00:00:27.230 --> 00:00:28.868
- [Voiceover] It's
basically everyone woke up

00:00:28.868 --> 00:00:29.796
one morning and were like

00:00:29.796 --> 00:00:30.934
"oh we're done with the Gilded Age,

00:00:30.934 --> 00:00:32.397
now it's time for the Progressive Era".

00:00:32.397 --> 00:00:34.278
- [Voiceover] Hooray,
Spanish-American wartimes!

00:00:34.278 --> 00:00:35.111
- [Voiceover] More first the

00:00:35.111 --> 00:00:36.251
Spanish-American War.

00:00:36.251 --> 00:00:38.597
So the Gilded Age is kind of this period

00:00:38.597 --> 00:00:41.818
of really intense industrialization

00:00:41.818 --> 00:00:45.860
and kind of focusing on
America's development

00:00:45.860 --> 00:00:48.268
as an industrial and business power.

00:00:48.268 --> 00:00:49.394
It's very inward looking,

00:00:49.394 --> 00:00:51.635
whereas after the Spanish-American war

00:00:51.635 --> 00:00:53.492
the United States takes a bigger role

00:00:53.492 --> 00:00:54.816
on the world's stage.

00:00:54.816 --> 00:00:57.739
The sarcastic part of the Gilded Age

00:00:57.739 --> 00:00:59.729
is that this was a term coined

00:00:59.729 --> 00:01:01.029
by Mark Twain

00:01:01.029 --> 00:01:02.196
of all people,

00:01:03.263 --> 00:01:04.096
in 1890.

00:01:06.064 --> 00:01:08.862
And he wrote a book
called "The Gilded Age",

00:01:08.862 --> 00:01:10.732
and what Twain was trying to say

00:01:10.732 --> 00:01:13.030
was that the United States in this period

00:01:13.030 --> 00:01:15.780
wasn't experiencing a golden age,

00:01:16.711 --> 00:01:20.310
an era of prosperity and happiness,

00:01:20.310 --> 00:01:22.144
but rather a Gilded Age.

00:01:22.144 --> 00:01:23.526
- [Voiceover] Oh so just like a thin layer

00:01:23.526 --> 00:01:24.726
of gold on top,

00:01:24.726 --> 00:01:26.951
disguising the cheap tin beneath?

00:01:26.951 --> 00:01:27.784
- [Voiceover] Exactly.

00:01:27.784 --> 00:01:29.061
- [Voiceover] Oh snap, Mark Twain.

00:01:29.061 --> 00:01:30.477
(laughter)

00:01:30.477 --> 00:01:31.722
So what's going on in this era

00:01:31.722 --> 00:01:33.441
that earns this nickname?

00:01:33.441 --> 00:01:35.524
What is the appearance of

00:01:36.648 --> 00:01:37.809
fancy,

00:01:37.809 --> 00:01:38.642
lovely gold

00:01:38.642 --> 00:01:39.841
that just turns out

00:01:39.841 --> 00:01:41.160
to be tacky and miserable?

00:01:41.160 --> 00:01:42.365
- [Voiceover] Well I think what people

00:01:42.365 --> 00:01:44.009
are talking about under the title

00:01:44.009 --> 00:01:45.109
of Gilded Age is that

00:01:45.109 --> 00:01:48.525
it's this time when immense wealth

00:01:48.525 --> 00:01:52.025
is accumulated by a number of individuals.

00:01:53.564 --> 00:01:54.841
Many of whom still have

00:01:54.841 --> 00:01:57.407
their names on things today.

00:01:57.407 --> 00:01:59.648
Like Andrew Carnegie,

00:01:59.648 --> 00:02:02.306
of Carnegie Mellon University,

00:02:02.306 --> 00:02:04.141
or Carnegie Hall.

00:02:04.141 --> 00:02:05.141
J.P. Morgan,

00:02:07.556 --> 00:02:09.896
who was a banking magnet.

00:02:09.896 --> 00:02:11.300
We still have J.P. Morgan

00:02:11.300 --> 00:02:13.437
as a financial institution today.

00:02:13.437 --> 00:02:17.094
And I would say John D.
Rockefeller is another.

00:02:17.094 --> 00:02:20.019
He was a founder of standard oil,

00:02:20.019 --> 00:02:21.993
so he was an oil baron.

00:02:21.993 --> 00:02:24.292
Of Rockefeller Center, right.

00:02:24.292 --> 00:02:25.476
So these are the individuals

00:02:25.476 --> 00:02:27.113
who got enormously wealthy

00:02:27.113 --> 00:02:28.518
in the Gilded Age.

00:02:28.518 --> 00:02:31.014
And they got wealthy by being the captians

00:02:31.014 --> 00:02:33.643
of these new expanding industries.

00:02:33.643 --> 00:02:34.811
- [Voiceover] Titans of industry.

00:02:34.811 --> 00:02:36.357
- [Voiceover] The titans of industry.

00:02:36.357 --> 00:02:37.190
So steel,

00:02:38.703 --> 00:02:39.703
and banking,

00:02:41.071 --> 00:02:42.731
and oil.

00:02:42.731 --> 00:02:44.984
But they got rich partly

00:02:44.984 --> 00:02:47.539
through political corruption.

00:02:47.539 --> 00:02:49.848
So one of the less than savory parts

00:02:49.848 --> 00:02:51.346
of the Gilded Age is that.

00:02:51.346 --> 00:02:52.844
A lot of this was done through

00:02:52.844 --> 00:02:54.992
political kickbacks,

00:02:54.992 --> 00:02:57.128
bribing of officials,

00:02:57.128 --> 00:02:58.892
bribing of the vice president

00:02:58.892 --> 00:02:59.903
of the United States.

00:02:59.903 --> 00:03:01.331
- [Voiceover] Woah woah woah, what?

00:03:01.331 --> 00:03:02.261
They bribed the vice president?

00:03:02.261 --> 00:03:03.661
(laughter)

00:03:03.661 --> 00:03:05.382
- [Voiceover] That was a scandal that

00:03:05.382 --> 00:03:07.925
dogged the presidency of

00:03:07.925 --> 00:03:09.005
Ulysses S. Grant

00:03:09.005 --> 00:03:09.968
- [Voiceover] Oh man.

00:03:09.968 --> 00:03:11.907
- [Voiceover] Credit Mobilier

00:03:11.907 --> 00:03:13.035
was a construction company

00:03:13.035 --> 00:03:15.535
that had less than savory ties

00:03:16.389 --> 00:03:19.349
to some people very high up in government.

00:03:19.349 --> 00:03:21.822
So these fortunes of people

00:03:21.822 --> 00:03:23.285
like Carnegie and Morgan

00:03:23.285 --> 00:03:26.543
were built partly on political corruption,

00:03:26.543 --> 00:03:28.460
and partly on the backs

00:03:30.118 --> 00:03:31.319
of waves

00:03:31.319 --> 00:03:33.683
of immigrant laborers.

00:03:33.683 --> 00:03:36.401
- [Voiceover] Like my ancestors.

00:03:36.401 --> 00:03:38.219
- [Voiceover] Like my ancestors.

00:03:38.219 --> 00:03:40.700
And so during this time period

00:03:40.700 --> 00:03:43.336
in the eastern part of Europe,

00:03:43.336 --> 00:03:44.775
in the southern part of Europe,

00:03:44.775 --> 00:03:47.870
there were political eruptions,

00:03:47.870 --> 00:03:50.302
there were just general poverty

00:03:50.302 --> 00:03:51.987
especially in Italy.

00:03:51.987 --> 00:03:54.717
In Russia there were a number of pogroms,

00:03:54.717 --> 00:03:57.743
which sent Russian Jews out of Russia

00:03:57.743 --> 00:04:00.774
and they came to the United States.

00:04:00.774 --> 00:04:03.560
And they came from very bad situations,

00:04:03.560 --> 00:04:05.278
so what they came to
was a little bit better

00:04:05.278 --> 00:04:06.776
but it wasn't much better.

00:04:06.776 --> 00:04:07.914
- [Voiceover] Which is
how they all ended up

00:04:07.914 --> 00:04:09.261
living in this one apartment

00:04:09.261 --> 00:04:10.782
in lower Manhattan,

00:04:10.782 --> 00:04:11.896
as you've detailed

00:04:11.896 --> 00:04:13.614
in this Jacob Riis photograph.

00:04:13.614 --> 00:04:14.578
- [Voiceover] Right.

00:04:14.578 --> 00:04:17.179
Okay so, Jacob Riss was

00:04:17.179 --> 00:04:20.244
a photographer in the 1890's.

00:04:20.244 --> 00:04:23.634
And he went around the lower east side

00:04:23.634 --> 00:04:25.282
of Manhattan,

00:04:25.282 --> 00:04:27.082
basically just photographing

00:04:27.082 --> 00:04:30.656
what he called "How the Other Half Lives".

00:04:30.656 --> 00:04:33.863
By the other half he meant immigrants,

00:04:33.863 --> 00:04:35.673
the poor who were living in the

00:04:35.673 --> 00:04:37.368
lower east side of Manhattan.

00:04:37.368 --> 00:04:38.971
Which, at that time,

00:04:38.971 --> 00:04:40.445
was the most dense

00:04:40.445 --> 00:04:42.862
section of humanity on Earth.

00:04:43.893 --> 00:04:45.157
And you can see here that

00:04:45.157 --> 00:04:47.495
this is a photograph from a tenement

00:04:47.495 --> 00:04:49.211
that he took a picture of.

00:04:49.211 --> 00:04:50.886
- [Voiceover] And what is a tenement?

00:04:50.886 --> 00:04:53.170
- [Voiceover] So tenements were these

00:04:53.170 --> 00:04:54.833
apartment dwellings

00:04:54.833 --> 00:04:56.219
which kind of sprang up,

00:04:56.219 --> 00:04:57.763
often in the backyards

00:04:57.763 --> 00:05:00.170
of other normal buildings.

00:05:00.170 --> 00:05:02.347
But they were sort of hastily constructed,

00:05:02.347 --> 00:05:05.160
they had many rooms in them,

00:05:05.160 --> 00:05:06.497
they weren't up to fire code.

00:05:06.497 --> 00:05:09.433
I'm not even sure there was a fire code.

00:05:09.433 --> 00:05:12.516
Most of the rooms didn't have windows

00:05:13.510 --> 00:05:15.624
or electric lighting,

00:05:15.624 --> 00:05:16.823
or ventilation.

00:05:16.823 --> 00:05:18.544
- [Voiceover] Sounds like
a great place to live.

00:05:18.544 --> 00:05:19.810
(laughter)

00:05:19.810 --> 00:05:21.678
But does it have curb appeal, Kim?

00:05:21.678 --> 00:05:22.642
(laughter)

00:05:22.642 --> 00:05:24.091
- [Voiceover] There is some pictures

00:05:24.091 --> 00:05:25.393
that can show you no,

00:05:25.393 --> 00:05:26.502
most of them were next

00:05:26.502 --> 00:05:28.174
to giant rotting heaps of garbage.

00:05:28.174 --> 00:05:29.126
- [Voiceover] Oh boy.

00:05:29.126 --> 00:05:31.183
- [Voiceover] One of my favorite things

00:05:31.183 --> 00:05:32.016
about the story of Jacob Riis

00:05:32.016 --> 00:05:33.902
is that he was a pioneer

00:05:33.902 --> 00:05:35.418
in the field of photography

00:05:35.418 --> 00:05:38.283
because he used flash photography.

00:05:38.283 --> 00:05:39.258
So you know,

00:05:39.258 --> 00:05:40.374
you've seen in old movies

00:05:40.374 --> 00:05:42.912
those flashbulbs that go off, right?

00:05:42.912 --> 00:05:44.254
Cause that was the only way

00:05:44.254 --> 00:05:45.846
he could get these apartments

00:05:45.846 --> 00:05:47.054
- [Voiceover] Cause there was

00:05:47.054 --> 00:05:47.887
no lighting in them.

00:05:47.887 --> 00:05:49.087
- [Voiceover] There was
no lighting in them.

00:05:49.087 --> 00:05:50.035
so you couldn't take a picture

00:05:50.035 --> 00:05:51.069
of them without light.

00:05:51.069 --> 00:05:53.549
So he brought his flash camera

00:05:53.549 --> 00:05:55.783
and he regularly set things on fire.

00:05:55.783 --> 00:05:56.616
- [Voiceover] Oh no.

00:05:56.616 --> 00:05:57.449
(laughter)

00:05:57.449 --> 00:05:58.432
- [Voiceover] In these apartments

00:05:58.432 --> 00:05:59.265
as he was trying to document

00:05:59.265 --> 00:06:00.098
what life was like.

00:06:00.098 --> 00:06:00.931
- [Voiceover] Oh that's awful.

00:06:00.931 --> 00:06:03.793
- [Voiceover] So this
is a photograph by Riis

00:06:03.793 --> 00:06:06.553
called "Five Cents a Spot",

00:06:06.553 --> 00:06:08.855
and so you paid five cents a night

00:06:08.855 --> 00:06:09.688
to live in this apartment.

00:06:09.688 --> 00:06:12.206
And so if you count here,

00:06:12.206 --> 00:06:14.039
there's one,

00:06:14.039 --> 00:06:15.072
two,

00:06:15.072 --> 00:06:16.305
three,

00:06:16.305 --> 00:06:17.686
four,

00:06:17.686 --> 00:06:18.708
five,

00:06:18.708 --> 00:06:19.541
six,

00:06:19.541 --> 00:06:21.343
I think this is somebody else's legs,

00:06:21.343 --> 00:06:23.593
seven, did I get everybody?

00:06:24.443 --> 00:06:26.266
There might be somebody
else hiding over here.

00:06:26.266 --> 00:06:28.707
Seven men sharing this room,

00:06:28.707 --> 00:06:31.267
and they're just doing the best they can.

00:06:31.267 --> 00:06:32.856
- [Voiceover] Right.

00:06:32.856 --> 00:06:34.841
- [Voiceover] And you
compare that with this,

00:06:34.841 --> 00:06:38.480
which is John D. Rockefeller's mansion

00:06:38.480 --> 00:06:39.931
in New York.

00:06:39.931 --> 00:06:41.081
It's called

00:06:41.081 --> 00:06:42.404
and I'm gonna butcher this

00:06:42.404 --> 00:06:43.237
"Kykuit".

00:06:45.632 --> 00:06:49.301
So the real question of the Gilded Age

00:06:49.301 --> 00:06:50.566
is how is it

00:06:50.566 --> 00:06:53.233
that some people get so wealthy,

00:06:54.246 --> 00:06:56.568
while some people are incredibly poor?

00:06:56.568 --> 00:06:57.648
- [Voiceover] Sure.

00:06:57.648 --> 00:07:00.318
- [Voiceover] And whose
responsibility is that?

00:07:00.318 --> 00:07:01.814
At one point,

00:07:01.814 --> 00:07:04.868
J.P. Morgan decides he is going

00:07:04.868 --> 00:07:07.551
to buy out Andrew Carnegie.

00:07:07.551 --> 00:07:09.002
He buys him out for more than

00:07:09.002 --> 00:07:11.301
four hundred million dollars,

00:07:11.301 --> 00:07:14.122
he loans money to the U.S. government.

00:07:14.122 --> 00:07:16.688
So he is the single biggest creditor

00:07:16.688 --> 00:07:18.407
to the U.S. government.

00:07:18.407 --> 00:07:20.171
At that point who has more power,

00:07:20.171 --> 00:07:21.309
the federal government,

00:07:21.309 --> 00:07:22.239
or J.P. Morgan?

00:07:22.239 --> 00:07:23.530
- [Voiceover] So we're really talking

00:07:23.530 --> 00:07:26.028
about the clash of two

00:07:26.028 --> 00:07:29.434
great and terrible energies, right?

00:07:29.434 --> 00:07:31.341
Like this immense wealth,

00:07:31.341 --> 00:07:33.184
and this immense deprivation.

00:07:33.184 --> 00:07:34.176
- [Voiceover] Yeah.

00:07:34.176 --> 00:07:35.555
- [Voiceover] How do they play out?

00:07:35.555 --> 00:07:36.991
- [Voiceover] We'll get to that

00:07:36.991 --> 00:07:37.824
in some more videos.


Gradient

WEBVTT
Kind: captions
Language: en

00:00:00.931 --> 00:00:02.913
- [Voiceover] So here I'm
gonna talk about the gradient.

00:00:02.913 --> 00:00:05.040
And in this video, I'm only gonna describe

00:00:05.040 --> 00:00:06.314
how you compute the gradient,

00:00:06.314 --> 00:00:07.614
and in the next couple ones

00:00:07.614 --> 00:00:09.705
I'm gonna give the
geometric interpretation.

00:00:09.705 --> 00:00:10.823
And I hate doing this,

00:00:10.823 --> 00:00:11.889
I hate showing the computation

00:00:11.889 --> 00:00:14.097
before the geometric intuition

00:00:14.097 --> 00:00:15.929
since usually it should
go the other way around,

00:00:15.929 --> 00:00:17.637
but the gradient is one
of those weird things

00:00:17.637 --> 00:00:19.732
where the way that you compute it

00:00:19.732 --> 00:00:22.383
actually seems kind of
unrelated to the intuition

00:00:22.383 --> 00:00:23.117
and you'll see that.

00:00:23.117 --> 00:00:25.028
We'll connect them in the next few videos.

00:00:25.028 --> 00:00:26.209
But to do that, we need to know

00:00:26.209 --> 00:00:28.167
what both of them actually are.

00:00:28.167 --> 00:00:30.002
So on the computation side of things,

00:00:30.002 --> 00:00:32.164
let's say you have some sort of function.

00:00:32.164 --> 00:00:34.236
And I'm just gonna make it
a two-variable function.

00:00:34.236 --> 00:00:39.110
And let's say it's f of x, y,
equals x-squared sine of y.

00:00:40.977 --> 00:00:42.857
The gradient is a way of packing together

00:00:42.857 --> 00:00:45.908
all the partial derivative
information of a function.

00:00:45.908 --> 00:00:48.070
So let's just start by computing
the partial derivatives

00:00:48.070 --> 00:00:48.972
of this guy.

00:00:48.972 --> 00:00:52.619
So partial of f with respect to x

00:00:53.972 --> 00:00:54.830
is equal to,

00:00:54.830 --> 00:00:57.071
so we look at this and we
consider x the variable

00:00:57.071 --> 00:00:58.659
and y the constant.

00:00:58.659 --> 00:01:01.424
Well in that case sine
of y is also a constant.

00:01:01.424 --> 00:01:03.211
As far as x is concerned,

00:01:03.211 --> 00:01:05.414
the derivative of x is 2x

00:01:05.414 --> 00:01:06.793
so we see that this will

00:01:06.793 --> 00:01:10.177
be 2x times that constant sine of y,

00:01:11.338 --> 00:01:13.148
sine of y.

00:01:13.148 --> 00:01:14.905
Whereas the partial derivative

00:01:14.905 --> 00:01:16.492
with respect to y.

00:01:19.016 --> 00:01:19.949
Now we look up here

00:01:19.949 --> 00:01:22.380
and we say x is considered a constant

00:01:22.380 --> 00:01:24.643
so x-squared is also considered a constant

00:01:24.643 --> 00:01:26.959
so this is just a
constant times sine of y,

00:01:26.959 --> 00:01:29.533
so that's gonna equal

00:01:29.533 --> 00:01:32.464
that same constant times the cosine of y,

00:01:32.464 --> 00:01:34.328
which is the derivative of sine.

00:01:36.049 --> 00:01:37.069
So now what the gradient does

00:01:37.069 --> 00:01:39.154
is it just puts both of these together

00:01:39.154 --> 00:01:40.132
in a vector.

00:01:40.132 --> 00:01:44.445
And specifically, maybe
I'll change colors here,

00:01:44.445 --> 00:01:48.127
you denote it with a little
upside-down triangle.

00:01:48.127 --> 00:01:49.958
The name of that symbol is nabla,

00:01:49.958 --> 00:01:51.871
but you often just pronounce it del,

00:01:51.871 --> 00:01:54.273
you'd say del f or gradient of f.

00:01:55.183 --> 00:01:57.215
And what this equals

00:01:57.215 --> 00:01:59.120
is a vector

00:01:59.120 --> 00:02:01.041
that has those two
partial derivatives in it.

00:02:01.041 --> 00:02:02.422
So the first one is the partial derivative

00:02:02.422 --> 00:02:03.651
with respect to x,

00:02:03.651 --> 00:02:06.083
to x times sine of y.

00:02:07.898 --> 00:02:11.045
And the bottom one, partial
derivative with respect to y

00:02:11.045 --> 00:02:14.558
X-squared cosine of y.

00:02:19.060 --> 00:02:20.432
And notice, maybe I should emphasize,

00:02:20.432 --> 00:02:23.242
this is actually a vector-valued function.

00:02:24.791 --> 00:02:28.050
So maybe I'll give it a
little bit more room here

00:02:28.050 --> 00:02:32.047
and emphasize that it's got an x and a y.

00:02:32.047 --> 00:02:33.907
This is a function that takes in

00:02:33.907 --> 00:02:35.767
a point in two-dimensional space

00:02:35.767 --> 00:02:38.371
and outputs a two-dimensional vector.

00:02:38.371 --> 00:02:40.616
So you could also imagine doing this

00:02:40.616 --> 00:02:42.552
with three different variables.

00:02:42.552 --> 00:02:44.350
Then you would have three
partial derivatives,

00:02:44.350 --> 00:02:47.076
and a three-dimensional output.

00:02:47.076 --> 00:02:49.467
And the way you might
write this more generally

00:02:49.467 --> 00:02:52.012
is we could go down here
and say the gradient

00:02:52.012 --> 00:02:53.807
of any function

00:02:53.807 --> 00:02:57.952
is equal to a vector with
its partial derivatives.

00:02:57.952 --> 00:03:01.126
Partial of f with respect to x,

00:03:02.633 --> 00:03:05.604
and partial of f with respect to y.

00:03:05.604 --> 00:03:09.060
And in some sense, we call
these partial derivatives.

00:03:09.060 --> 00:03:11.311
I like to think as the
gradient as the full derivative

00:03:11.311 --> 00:03:12.977
cuz it kind of captures
all of the information

00:03:12.977 --> 00:03:14.324
that you need.

00:03:14.324 --> 00:03:15.825
So a very helpful mnemonic device

00:03:15.825 --> 00:03:18.388
with the gradient is to
think about this triangle,

00:03:18.388 --> 00:03:21.158
this nabla symbol as being a vector

00:03:21.158 --> 00:03:25.151
full of partial derivative operators.

00:03:25.151 --> 00:03:28.465
And by operator, I just mean like partial

00:03:28.465 --> 00:03:29.644
with respect to x,

00:03:29.644 --> 00:03:31.267
something where you
could give it a function,

00:03:31.267 --> 00:03:32.660
and it gives you another function.

00:03:32.660 --> 00:03:34.649
So you give this guy the function f

00:03:34.649 --> 00:03:36.516
and it gives you this expression,

00:03:36.516 --> 00:03:38.790
this multi-variable function as a result.

00:03:38.790 --> 00:03:41.595
So the nabla symbol is this vector full

00:03:41.595 --> 00:03:44.229
of different partial derivative operators.

00:03:44.229 --> 00:03:46.749
And in this case it might
just be two of them,

00:03:46.749 --> 00:03:47.932
and this is kind of a weird thing

00:03:47.932 --> 00:03:49.233
because it's like what,

00:03:49.233 --> 00:03:51.140
this is a vector, it's
got like operators in it,

00:03:51.140 --> 00:03:52.529
that's not what I thought vectors do.

00:03:52.529 --> 00:03:54.844
But you can kind of see where it's going.

00:03:54.844 --> 00:03:55.907
It's really just,

00:03:55.907 --> 00:03:57.899
you can think of it as a memory trick,

00:03:57.899 --> 00:04:00.464
but in some sense it's a
little bit deeper than that.

00:04:00.464 --> 00:04:02.609
And really when you take this triangle

00:04:02.609 --> 00:04:04.325
and you say ok let's take this triangle

00:04:04.325 --> 00:04:07.168
and you can kind of imagine
multiplying it by f,

00:04:07.168 --> 00:04:09.280
really it's like an operator
taking in this function

00:04:09.280 --> 00:04:11.476
and it's gonna give you another function.

00:04:11.476 --> 00:04:13.105
It's like you take this
triangle and you put an f

00:04:13.105 --> 00:04:15.221
in front of it, and you can imagine,

00:04:15.221 --> 00:04:17.337
like this part gets
multipled, quote unquote

00:04:17.337 --> 00:04:19.741
multiplied with f, this
part gets quote unquote

00:04:19.741 --> 00:04:21.410
multiplied with f but
really you're just saying

00:04:21.410 --> 00:04:24.168
you take the partial
derivative with respect to x

00:04:24.168 --> 00:04:26.073
and then with y, and on and on.

00:04:26.073 --> 00:04:27.178
And the reason for doing this,

00:04:27.178 --> 00:04:29.536
this symbol comes up a
lot in other contexts.

00:04:29.536 --> 00:04:31.775
There are two other operators
that you're gonna learn about

00:04:31.775 --> 00:04:33.613
called the divergence and the curl.

00:04:33.613 --> 00:04:34.507
We'll get to those later,

00:04:34.507 --> 00:04:35.572
all in due time.

00:04:35.572 --> 00:04:37.083
But it's useful to think about this

00:04:37.083 --> 00:04:40.296
vector-ish thing of partial derivatives.

00:04:40.296 --> 00:04:42.898
And I mean one weird thing about it,

00:04:42.898 --> 00:04:45.869
you could say ok so this
nabla symbol is a vector

00:04:45.869 --> 00:04:47.050
of partial derivative operators.

00:04:47.050 --> 00:04:48.923
What's its dimension?

00:04:48.923 --> 00:04:51.584
And it's like how many
dimensions do you got?

00:04:51.584 --> 00:04:53.521
Because if you had a
three-dimensional function

00:04:53.521 --> 00:04:55.032
that would mean that you should treat this

00:04:55.032 --> 00:04:57.840
like it's got three different
operators as part of it.

00:04:57.840 --> 00:05:01.708
And you know I'd kinda,
finish this off down here,

00:05:01.708 --> 00:05:03.864
and if you had something
that was 100-dimensional

00:05:03.864 --> 00:05:06.513
it would have 100
different operators in it

00:05:06.513 --> 00:05:07.489
and that's fine.

00:05:07.489 --> 00:05:08.426
It's really just again,

00:05:08.426 --> 00:05:10.136
kind of a memory trick.

00:05:10.136 --> 00:05:13.438
So with that, that's how
you compute the gradient.

00:05:13.438 --> 00:05:14.241
Not too much too it,

00:05:14.241 --> 00:05:15.673
it's pretty much just partial derivatives,

00:05:15.673 --> 00:05:17.213
but you smack em into a vector

00:05:17.213 --> 00:05:19.292
where it gets fun and
where it gets interesting

00:05:19.292 --> 00:05:21.612
is with the geometric interpretation.

00:05:21.612 --> 00:05:22.875
I'll get to that in
the next couple videos.

00:05:22.875 --> 00:05:24.949
It's also a super important tool

00:05:24.949 --> 00:05:27.232
for something called the
directional derivative.

00:05:27.232 --> 00:05:29.483
So you've got a lot of fun stuff ahead.


Ideal circuit elements

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:01.878
- [Voiceover] We're now
ready to start the study

00:00:01.878 --> 00:00:04.734
of circuit analysis and to design circuits

00:00:04.734 --> 00:00:06.755
and analyze circuits, one
of the things we need to do

00:00:06.755 --> 00:00:08.983
is have something to build circuits with

00:00:08.983 --> 00:00:10.957
and that's what we're gonna
talk about in this video.

00:00:10.957 --> 00:00:14.138
The idea is we're gonna
have three circuit elements.

00:00:14.138 --> 00:00:16.344
These circuit elements
are called resistor,

00:00:21.594 --> 00:00:22.458
capacitor,

00:00:26.958 --> 00:00:29.369
i-tor,

00:00:29.369 --> 00:00:30.713
and inductor.

00:00:37.663 --> 00:00:40.050
Okay, these are the three passive

00:00:40.050 --> 00:00:42.592
or two-element components

00:00:42.592 --> 00:00:44.592
or circuit elements that
we're gonna use to design

00:00:44.592 --> 00:00:46.647
a lot of different kinds of circuits.

00:00:46.647 --> 00:00:49.874
First, I want to introduce a
symbol for each one of these

00:00:49.874 --> 00:00:52.707
so we can talk about it
and making drawings of it

00:00:52.707 --> 00:00:54.854
and first is gonna be the resistor.

00:00:57.614 --> 00:00:59.131
Resistor symbol looks like this.

00:00:59.131 --> 00:01:01.604
It's a zigzag line like that

00:01:01.604 --> 00:01:05.284
representing current going
through and being resisted

00:01:05.284 --> 00:01:06.771
having to do some work.

00:01:06.771 --> 00:01:09.372
Another symbol for
resistor looks like this

00:01:09.372 --> 00:01:11.101
used in other parts of the world

00:01:11.101 --> 00:01:13.121
besides the United States and Japan.

00:01:13.121 --> 00:01:14.700
That's what a resistor looks like

00:01:14.700 --> 00:01:17.080
and the symbol we use is R.

00:01:17.080 --> 00:01:19.739
Now for the capacitor,

00:01:19.739 --> 00:01:22.502
capacitor symbol is
actually a capacitor's built

00:01:22.502 --> 00:01:25.884
from two conductors or metal objects

00:01:25.884 --> 00:01:27.589
that are placed close together

00:01:27.589 --> 00:01:30.145
and most capacitors sort of look like that

00:01:30.145 --> 00:01:31.782
when they're actually built

00:01:31.782 --> 00:01:34.230
and the symbol for capacitor is a C.

00:01:34.230 --> 00:01:36.309
And finally for the inductor,

00:01:37.619 --> 00:01:39.175
we'll do inductors like this.

00:01:39.175 --> 00:01:42.809
An inductor is actually
built from a coil of wire

00:01:42.809 --> 00:01:44.702
and so when we draw an inductor symbol,

00:01:44.702 --> 00:01:48.009
we draw a little coil of wire like that

00:01:48.009 --> 00:01:52.143
and the symbol is L which is a little odd.

00:01:52.143 --> 00:01:55.267
It could be called i but the symbol for i

00:01:55.267 --> 00:01:57.670
was already taken by current

00:01:57.670 --> 00:02:00.770
which is from the French for intensity

00:02:00.770 --> 00:02:02.732
and we couldn't use C for current

00:02:02.732 --> 00:02:05.053
because the C is used here

00:02:05.053 --> 00:02:07.898
so it's a little quirk
of our nomenclature.

00:02:07.898 --> 00:02:08.741
All right.

00:02:08.741 --> 00:02:12.827
Each of these components has an equation

00:02:12.827 --> 00:02:13.919
that goes along with it

00:02:13.919 --> 00:02:16.264
that relates the voltage to the current.

00:02:16.264 --> 00:02:17.506
Now, I'm gonna go back here.

00:02:17.506 --> 00:02:19.990
I'm gonna label the voltages
and currents on here

00:02:19.990 --> 00:02:24.990
in a very important convention
for drawing circuits.

00:02:25.007 --> 00:02:26.411
Let's do that.

00:02:26.411 --> 00:02:28.989
When we talk about the
voltage on a component,

00:02:28.989 --> 00:02:33.111
we can label it however
we want, plus-minus V,

00:02:33.111 --> 00:02:37.105
and we draw the current going in.

00:02:37.105 --> 00:02:39.043
I'll just label a little i there

00:02:39.043 --> 00:02:40.634
and we'll do it on all these.

00:02:40.634 --> 00:02:43.851
The current goes into
the positive terminal.

00:02:47.301 --> 00:02:49.711
The current goes into
the positive terminal

00:02:49.711 --> 00:02:51.777
so that's a V on the capacitor

00:02:55.507 --> 00:02:56.433
and finally,

00:03:01.323 --> 00:03:02.987
and the current goes in

00:03:02.987 --> 00:03:05.448
and we're gonna be very
consistent about this

00:03:05.448 --> 00:03:07.723
and that's gonna keep
us from making mistakes.

00:03:07.723 --> 00:03:09.941
All right, so let's go
back to our resistor

00:03:09.941 --> 00:03:11.822
and we're gonna do the
equation for a resistor.

00:03:11.822 --> 00:03:15.432
What is the I-V equation for a resistor?

00:03:15.432 --> 00:03:18.509
I-V equation means what
relates current to voltage

00:03:18.509 --> 00:03:21.911
and for a resistor, it's V

00:03:21.911 --> 00:03:26.211
equals i times R

00:03:27.191 --> 00:03:29.604
so the voltage across the resistor

00:03:29.604 --> 00:03:33.795
is equal to the current
through the resistor

00:03:33.795 --> 00:03:37.313
times this constant of proportionality

00:03:37.313 --> 00:03:38.872
that we call the resistance.

00:03:38.872 --> 00:03:40.659
This has a very important name.

00:03:40.659 --> 00:03:45.512
This is called Ohm's law

00:03:45.512 --> 00:03:47.683
and you're gonna use this a lot

00:03:49.123 --> 00:03:51.198
so that's Ohm's law right there.

00:03:51.198 --> 00:03:53.404
This is Ohm's law.

00:03:56.394 --> 00:03:59.622
Now for the IV relationship
for the capacitor,

00:03:59.622 --> 00:04:02.127
the capacitor has that
property that the current

00:04:02.127 --> 00:04:06.388
through the capacitor is
proportional to the rate of change

00:04:06.388 --> 00:04:09.348
of the voltage, not to the
voltage but to the rate of change

00:04:09.348 --> 00:04:14.348
of the voltage and the way we
write that is current equals,

00:04:15.827 --> 00:04:18.441
C is the proportionality constant,

00:04:18.441 --> 00:04:22.087
and we write dv, dt

00:04:23.616 --> 00:04:25.583
so this is the rate of change of voltage

00:04:25.583 --> 00:04:27.243
with respect to time.

00:04:27.243 --> 00:04:29.681
We multiply that by this
property of this device

00:04:29.681 --> 00:04:33.745
called capacitance and
that gives us the current.

00:04:33.745 --> 00:04:35.882
This doesn't have a special name

00:04:35.882 --> 00:04:38.900
but I'm gonna refer to it
as the capacitor equation

00:04:40.450 --> 00:04:41.689
so now we have two equations.

00:04:41.689 --> 00:04:44.290
Let's do the third equation
which is for the inductor.

00:04:44.290 --> 00:04:47.657
The inductor has the property
very similar to the capacitor.

00:04:47.657 --> 00:04:50.570
It has the property
that the voltage across

00:04:50.570 --> 00:04:55.179
is proportional to the time
rate of change of the current

00:04:55.179 --> 00:04:57.119
flowing through the inductor

00:04:57.119 --> 00:05:00.462
so this is a similar but opposite
of how a capacitor works.

00:05:00.462 --> 00:05:04.583
The voltage is proportional
to the time rate of change

00:05:04.583 --> 00:05:06.882
of current and the way we write that

00:05:06.882 --> 00:05:11.192
is voltage equals L,

00:05:12.502 --> 00:05:16.190
di, dt.

00:05:20.200 --> 00:05:21.736
The voltage is proportional.

00:05:21.736 --> 00:05:24.371
The proportionality
constant is the inductants.

00:05:24.371 --> 00:05:26.392
The inductance of the inductor

00:05:26.392 --> 00:05:30.409
and this is the time rate
of change of voltage,

00:05:31.739 --> 00:05:34.480
OH sorry, the time rate
of change of current

00:05:34.480 --> 00:05:36.721
flowing through the inductor

00:05:37.901 --> 00:05:39.487
so this gives us our three equations.

00:05:39.487 --> 00:05:40.624
Here they are.

00:05:40.624 --> 00:05:42.307
These are three element equations

00:05:42.307 --> 00:05:45.686
and we're gonna use these all the time,

00:05:45.686 --> 00:05:47.788
right there, those three equations.

00:05:49.108 --> 00:05:50.381
One final point I wanna make

00:05:50.381 --> 00:05:53.760
is for both these equations of components,

00:05:53.760 --> 00:05:57.336
these are ideal, ideal components.

00:06:01.966 --> 00:06:05.800
That means these things are
mathematical perfect things

00:06:05.800 --> 00:06:08.554
that we have in our minds
that we're gonna try to build

00:06:08.554 --> 00:06:10.853
in the real world but we'll come close.

00:06:10.853 --> 00:06:12.131
We'll come very close.

00:06:12.131 --> 00:06:14.267
We now have a wonderful set of equations:

00:06:14.267 --> 00:06:16.148
V equals iR,

00:06:16.148 --> 00:06:18.365
i equals C, dv, dt.

00:06:18.365 --> 00:06:21.071
v equals L, di, dt.

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These are gonna be like
poetry for you pretty soon

00:06:23.334 --> 00:06:26.109
and these ideal equations will produce

00:06:26.109 --> 00:06:28.628
all kinds of really cool circuits for us.