Divergence intuition, part 2

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Hey everyone.

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So, in the last video I was
talking about divergence

00:00:03.560 --> 00:00:06.480
and kind of laying down the
intuition that we need for it.

00:00:06.480 --> 00:00:08.400
Where you're imagining a
vector field as representing

00:00:08.400 --> 00:00:10.900
some kind of fluid flow
where particles move

00:00:10.900 --> 00:00:12.920
according to the vector
that they're attached to

00:00:12.920 --> 00:00:15.420
in that point in time and as they move

00:00:15.420 --> 00:00:17.060
to a different point the
vector they're attached to

00:00:17.060 --> 00:00:20.300
is different so their
velocity changes in some way.

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And the key question that
we want to think about is

00:00:22.241 --> 00:00:24.200
"If you have a given
point somewhere in space,

00:00:24.200 --> 00:00:26.500
"does fluid tend to
flow towards that point

00:00:26.500 --> 00:00:28.200
"or does it tend to
flow more away from it?"

00:00:28.200 --> 00:00:30.580
Does it diverge away from that point?

00:00:30.580 --> 00:00:32.940
And what I wanna do here
is start kinda closing

00:00:32.940 --> 00:00:35.400
our grasp on that intuition
a little bit more tightly,

00:00:35.400 --> 00:00:38.300
as if we are trying to
discover the formula for

00:00:38.300 --> 00:00:41.140
divergence ourselves, because
ultimately that's what

00:00:41.140 --> 00:00:43.760
I'm gonna get to, a
formula for divergence.

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But I want it to be
something that's not just

00:00:45.340 --> 00:00:47.100
plopped down in front of
you, but something that you

00:00:47.100 --> 00:00:50.640
actually, you know,
feel deep in your bones.

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So a vector field like the
one I have pictured above

00:00:52.520 --> 00:00:55.601
is given as a function, a
multi-variable function,

00:00:55.601 --> 00:00:57.041
with a two-dimensional input,

00:00:57.041 --> 00:00:59.360
since it's a two-dimensional vector field,

00:00:59.360 --> 00:01:01.841
and then some kinda
two-dimensional output.

00:01:01.841 --> 00:01:05.924
And it's common to write
P and Q as the functions

00:01:07.880 --> 00:01:10.601
for these, the components of the output.

00:01:10.601 --> 00:01:12.960
So P and Q are each just
scalar value functions

00:01:12.960 --> 00:01:14.360
and you think of them as the components of

00:01:14.360 --> 00:01:15.940
your vector valued output.

00:01:15.940 --> 00:01:18.860
And the divergence is
kinda like a derivative,

00:01:18.860 --> 00:01:21.540
where you might denote it by just div,

00:01:21.540 --> 00:01:23.900
and in the same way that your
derivative, you have this

00:01:23.900 --> 00:01:26.241
operator and what it does
is it takes in a function.

00:01:26.241 --> 00:01:27.880
And what you get is a whole new function.

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This div operator you
think of as taking in

00:01:31.200 --> 00:01:34.320
a vector field of some kind
and you get a new function.

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And the new function you
get will be scalar valued,

00:01:36.681 --> 00:01:38.780
it'll be something that just
takes in points in space

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and outputs a number,
because what you're thinking,

00:01:41.220 --> 00:01:42.740
the thing that it's trying to do

00:01:42.740 --> 00:01:45.780
is take in some specific
point with XY coordinates

00:01:45.780 --> 00:01:48.020
and just give you a
single number to tell you

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"Hey, does fluid tend
to diverge away from it?

00:01:49.820 --> 00:01:54.200
"How much, or does it tend to
flow towards it and how much?"

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So this is the kind of, the form

00:01:56.480 --> 00:01:58.381
of the thing that we're going for.

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So here what we're gonna
do is just start thinking

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about cases where this divergence
is positive, or negative,

00:02:03.561 --> 00:02:05.561
or zero and what that should look like.

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So for example, let's say we
want cases where the divergence

00:02:08.881 --> 00:02:13.048
of our vector field at a
specific point XY is positive.

00:02:15.461 --> 00:02:17.341
What might that look like?

00:02:17.341 --> 00:02:20.121
So one case would be where your
point, nothing is happening

00:02:20.121 --> 00:02:22.301
at that point and the vector
attached to it is zero,

00:02:22.301 --> 00:02:25.241
and everyone around it is going away.

00:02:25.241 --> 00:02:28.342
This is kinda the extreme
example of positive divergence.

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And I animated this in the
last video where we have

00:02:31.222 --> 00:02:33.921
you know, all of the vectors
pointing away from the origin

00:02:33.921 --> 00:02:35.901
and if you look at a
region around that origin,

00:02:35.901 --> 00:02:38.421
all the fluid particles
kinda go out of that region.

00:02:38.421 --> 00:02:42.841
And that's the quintessential
positive divergence example.

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But it doesn't have to look like that.

00:02:44.061 --> 00:02:46.621
It actually, I mean, you
could have something where

00:02:46.621 --> 00:02:50.001
there is a little bit of
movement at your point

00:02:50.001 --> 00:02:52.381
and then maybe there's
movement towards it as well

00:02:52.381 --> 00:02:55.201
from one side, and vectors
are kind of going towards it,

00:02:55.201 --> 00:02:58.901
but they're going away
from it even more rapidly

00:02:58.901 --> 00:03:01.261
on the other side, so if
you think of any kind of

00:03:01.261 --> 00:03:03.462
actual region around your
point, you're saying,

00:03:03.462 --> 00:03:05.821
"Sure, fluid is going into
that region a little bit,

00:03:05.821 --> 00:03:07.701
"but it's much more
counterbalanced by how quickly

00:03:07.701 --> 00:03:08.921
it's going out."

00:03:08.921 --> 00:03:11.201
So these are the kind of
situations you might see

00:03:11.201 --> 00:03:13.041
for positive divergence.

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Now negative divergence,
negative divergence.

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Let's think about what examples
of that might look like.

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Divergence of V at a given point

00:03:21.103 --> 00:03:22.703
and you know, really it's
something that takes in

00:03:22.703 --> 00:03:24.064
all points of the plane
but we're just looking

00:03:24.064 --> 00:03:28.684
at specific points, so if
the divergence is negative,

00:03:28.684 --> 00:03:30.344
well the quintessential
example here is that

00:03:30.344 --> 00:03:32.604
nothing happens at your
point, but all of the vectors

00:03:32.604 --> 00:03:35.724
around it are kind of
flowing in towards it.

00:03:35.724 --> 00:03:36.985
And this is the thing where I animated,

00:03:36.985 --> 00:03:39.544
where we took this and we
flipped all of the vectors.

00:03:39.544 --> 00:03:41.624
And said, "Ah, there, if you
start playing the fluid flow,

00:03:41.624 --> 00:03:44.084
"then the density in any
region around the origin

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"you know, increases a lot,
all of the fluid particles

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"tend to converge towards that center."

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But again, this isn't the only
example that you might have.

00:03:51.464 --> 00:03:55.345
You could have a little bit of
activity at your point itself

00:03:55.345 --> 00:03:57.584
and maybe it is the case that
things do flow away from it

00:03:57.584 --> 00:03:59.065
a little bit as you're going away.

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And some of the fluid
particles are going away,

00:04:00.904 --> 00:04:03.025
and it's just the case
that the fluid particles

00:04:03.025 --> 00:04:05.104
flowing in towards it
from another direction

00:04:05.104 --> 00:04:07.024
heavily counterbalance that.

00:04:07.024 --> 00:04:08.824
Cuz then if you're looking
at any kind of region

00:04:08.824 --> 00:04:11.384
around your point, you say
fluid particles are coming in

00:04:11.384 --> 00:04:13.764
quite rapidly, a lot
of particles per time,

00:04:13.764 --> 00:04:16.244
but they're not leaving too
rapidly round the other end.

00:04:16.244 --> 00:04:18.564
So kind of loosely,
intuitively, this is what a

00:04:18.564 --> 00:04:21.224
negative divergence case might look like.

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And finally, another case that
we wanna start thinking about

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as we're tightening our
grasp on this intuition

00:04:26.305 --> 00:04:28.604
is what happens, or
what does it look like,

00:04:28.604 --> 00:04:32.024
if the divergence of your
function at a specific point

00:04:32.024 --> 00:04:36.064
is zero, right, if it's
just absolutely zero?

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And one thing this could
look like is, you know,

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you have something going on
but nothing really changes

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and all of the fluid just kinda
flows in then it flows out

00:04:44.344 --> 00:04:46.761
and on the whole it balances.

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You know, if you take any kind of region

00:04:48.705 --> 00:04:49.885
the amount flowing in is balanced

00:04:49.885 --> 00:04:51.785
with the amount flowing out.

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But it could also look like
you have fluid flowing in

00:04:54.705 --> 00:04:58.346
kind of from one dimension,
but it's cancelled out

00:04:58.346 --> 00:05:01.185
by flowing away from the point in a manner

00:05:01.185 --> 00:05:04.005
that sort of perfectly balances
it in another direction.

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So these are the loose pictures
that I want you to have

00:05:05.625 --> 00:05:08.066
in the back of your
mind as we start looking

00:05:08.066 --> 00:05:10.545
for the actual formula for divergence.

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And what I'll do in the next video or two,

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is start looking at
these functions P and Q

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and thinking about the
partial derivative properties

00:05:16.785 --> 00:05:19.285
that they have that will
correspond with, you know,

00:05:19.285 --> 00:05:21.905
these positive divergence
images that you should have

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in your head, or the
negative divergence images

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that you should have in your head.

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So with that, I'll see you next video.


Divergence notation

WEBVTT
Kind: captions
Language: en

00:00:00.172 --> 00:00:02.282
- [Voiceover] So I've said that
if you have a vector field,

00:00:02.282 --> 00:00:03.559
a two-dimensional vector field

00:00:03.559 --> 00:00:05.741
with component functions P and Q,

00:00:05.741 --> 00:00:09.793
that the divergence of this
guy, the divergence of v,

00:00:09.793 --> 00:00:13.300
which is a scalar-valued
function of x and y,

00:00:13.300 --> 00:00:16.957
is by definition, the
partial derivative of P,

00:00:16.957 --> 00:00:20.696
with respect to x, plus the
partial derivative of Q,

00:00:20.696 --> 00:00:22.320
with respect to y.

00:00:22.320 --> 00:00:24.282
And there's actually another
notation for divergence

00:00:24.282 --> 00:00:26.308
that's kind of helpful for
remembering the formula.

00:00:26.308 --> 00:00:29.159
And what it is, is you
take this nabla symbol,

00:00:29.159 --> 00:00:31.794
that upside down triangle that
we also use for the gradient,

00:00:31.794 --> 00:00:34.720
and imagine taking the
dot product between that

00:00:34.720 --> 00:00:36.487
and your vector-valued function.

00:00:36.487 --> 00:00:38.134
And as we did with the gradient,

00:00:38.134 --> 00:00:41.285
the loose mnemonic you have
for this upside down triangle,

00:00:41.285 --> 00:00:42.951
as you think of it as a vector,

00:00:42.951 --> 00:00:45.053
full of partial differential operators,

00:00:45.053 --> 00:00:47.014
and that sounds fancy but all it means

00:00:47.014 --> 00:00:49.453
is you kind of take
this partial, partial x,

00:00:49.453 --> 00:00:51.241
a thing that wants to take in a function

00:00:51.241 --> 00:00:52.706
and take its partial derivative,

00:00:52.706 --> 00:00:54.295
and that's its first component,

00:00:54.295 --> 00:00:57.591
and the second component
is this partial, partial y,

00:00:57.591 --> 00:00:59.356
a thing that wants to take in a function

00:00:59.356 --> 00:01:01.736
and take its partial
derivative with respect to y.

00:01:01.736 --> 00:01:03.718
And, you know, loosely
this isn't really a vector,

00:01:03.718 --> 00:01:06.276
these aren't numbers, or
functions, or things like that,

00:01:06.276 --> 00:01:07.695
but it's something you can write down

00:01:07.695 --> 00:01:10.014
and it'll be kind of helpful symbolically.

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And you imagine taking
the dot product with that,

00:01:12.557 --> 00:01:15.673
and, you know, v who has components,

00:01:15.673 --> 00:01:19.840
these scalar-valued functions,
P of xy, and Q of xy.

00:01:22.274 --> 00:01:24.622
And when you imagine
doing this dot product,

00:01:24.622 --> 00:01:25.861
and you're kind of lining up terms

00:01:25.861 --> 00:01:27.728
and the first one multiplied
by the second, right,

00:01:27.728 --> 00:01:30.198
quote unquote multiplied,
because, in this case,

00:01:30.198 --> 00:01:32.464
when I say this first
component multiplied by p,

00:01:32.464 --> 00:01:35.148
I really mean you're taking
that partial derivative operator

00:01:35.148 --> 00:01:39.070
partial, partial x,
and evaluating it at p.

00:01:39.070 --> 00:01:41.577
That's kind of what multiplication
looks like in this case.

00:01:41.577 --> 00:01:44.537
So, you take that, and as per
the dot product you then add,

00:01:44.537 --> 00:01:46.693
what happens if you take
this partial operator,

00:01:46.693 --> 00:01:49.939
this partial, partial
y, and quote unquote,

00:01:49.939 --> 00:01:51.250
multiply it with q.

00:01:51.250 --> 00:01:52.436
Which, in the case of an operator,

00:01:52.436 --> 00:01:54.200
means you kind of give it the function q

00:01:54.200 --> 00:01:56.213
and it's gonna take
its partial derivative.

00:01:56.213 --> 00:01:58.199
So, we see we get the
same thing over here,

00:01:58.199 --> 00:02:00.429
it's the same formula that we have,

00:02:00.429 --> 00:02:02.786
and it's just kind of a nice, little,

00:02:02.786 --> 00:02:04.545
you can think of it as a mnemonic device

00:02:04.545 --> 00:02:07.736
for remembering what the divergence is.

00:02:07.736 --> 00:02:11.819
But another nice thing, this can apply

00:02:11.819 --> 00:02:13.809
to higher-dimensional functions, as well.

00:02:13.809 --> 00:02:18.260
Right? If we have
something that, let's see,

00:02:18.260 --> 00:02:19.880
something that's a vector-valued function,

00:02:19.880 --> 00:02:21.867
and it's gonna be a
three-dimensional vector field.

00:02:21.867 --> 00:02:24.271
So, it's got x, y, and z as its inputs,

00:02:24.271 --> 00:02:27.027
and its output then also has
to have three dimensions.

00:02:27.027 --> 00:02:30.474
So, it might be like, P, Q, and R,

00:02:30.474 --> 00:02:32.682
and all of these are functions of x and y.

00:02:32.682 --> 00:02:36.698
So, that's P of x and y, Q,
oh no, x, y, and z, right?

00:02:36.698 --> 00:02:39.708
So, P of x, y, kind of got in the habit

00:02:39.708 --> 00:02:43.875
of two dimensions there, P of
x, y and z, Q of x, y, and z,

00:02:46.592 --> 00:02:49.120
and then R of x, y, and z.

00:02:49.120 --> 00:02:51.498
And I haven't talked about
three-dimensional divergence.

00:02:51.498 --> 00:02:52.581
But if you think of this

00:02:52.581 --> 00:02:55.929
and then you imagine
doing your nabla, dotted

00:02:55.929 --> 00:02:58.971
with the vector-valued function,
it can still make sense.

00:02:58.971 --> 00:03:01.654
And in this case, that
nabla you're thinking of

00:03:01.654 --> 00:03:04.002
as having three different components.

00:03:04.002 --> 00:03:07.878
It's gonna be, on the one
hand this partial, partial x,

00:03:07.878 --> 00:03:11.143
I should say partial x there, partial x,

00:03:11.143 --> 00:03:14.799
now the second component
is partial, partial y,

00:03:14.799 --> 00:03:16.790
and the last component
is partial, partial z.

00:03:16.790 --> 00:03:19.342
And the ordering of
these, of the variables,

00:03:19.342 --> 00:03:21.741
here, x, y and z is just
whatever I have here.

00:03:21.741 --> 00:03:23.981
So, even if they didn't
have the names x, y, z,

00:03:23.981 --> 00:03:24.994
you kinda out them in the same order

00:03:24.994 --> 00:03:27.059
that they show up in your function.

00:03:27.059 --> 00:03:30.055
And when you imagine taking
the dot product between this,

00:03:30.055 --> 00:03:32.964
and your P as a function, Q as a function,

00:03:32.964 --> 00:03:36.464
and R as a function, vector-valued output,

00:03:38.230 --> 00:03:41.729
what you get, and I'll write it over here,

00:03:41.729 --> 00:03:46.076
you take that partial, partial
x and kind of multiply it,

00:03:46.076 --> 00:03:49.757
with P, which means you're
really evaluating at P.

00:03:49.757 --> 00:03:51.840
So, partial x here.

00:03:51.840 --> 00:03:55.319
Then you add partial, partial y.

00:03:55.319 --> 00:03:57.977
And you're evaluating at Q,
because you're kind of imagining

00:03:57.977 --> 00:03:59.592
multiplying these second components.

00:03:59.592 --> 00:04:00.660
And you'll add what happens

00:04:00.660 --> 00:04:03.479
when you multiply by
these third components,

00:04:03.479 --> 00:04:07.646
or that's partial, partial
z, by that last component.

00:04:09.319 --> 00:04:10.990
And, you know, since I haven't talked

00:04:10.990 --> 00:04:12.321
about three-dimensional vector fields,

00:04:12.321 --> 00:04:15.046
with three-dimensional
divergence, this last term,

00:04:15.046 --> 00:04:17.308
maybe it's not given that you'd
have as strong an intuition

00:04:17.308 --> 00:04:19.690
for why this shows up in
divergence as the other two,

00:04:19.690 --> 00:04:20.803
but it's actually quite similar,

00:04:20.803 --> 00:04:23.561
you're thinking about changes
to the z component of a vector

00:04:23.561 --> 00:04:25.661
as the value z, of the input,

00:04:25.661 --> 00:04:26.643
as you're kind of moving up

00:04:26.643 --> 00:04:29.242
and down and that direction changes.

00:04:29.242 --> 00:04:31.652
But this pattern will go
for even higher dimensions

00:04:31.652 --> 00:04:34.694
that we can't visualize, four,
five, 100, whatever you want.

00:04:34.694 --> 00:04:37.100
And that's what makes this
notation here quite nice,

00:04:37.100 --> 00:04:38.643
is that it encapsulates that

00:04:38.643 --> 00:04:40.825
and gives a really compact way

00:04:40.825 --> 00:04:43.015
of describing this formula that,

00:04:43.015 --> 00:04:44.184
it has a simple pattern to it,

00:04:44.184 --> 00:04:47.137
but would otherwise kind of
get out of hand to write.

00:04:47.137 --> 00:04:47.970
See you next video.


Divergence formula, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.194 --> 00:00:01.515
- [Voiceover] Hello
again, in the last video

00:00:01.515 --> 00:00:02.795
we were looking at vector fields that

00:00:02.795 --> 00:00:04.382
only have an X component,

00:00:04.382 --> 00:00:05.742
basically meaning all of the vectors

00:00:05.742 --> 00:00:07.854
point just purely to the
left or to the right,

00:00:07.854 --> 00:00:09.970
with nothing up and down going on.

00:00:09.970 --> 00:00:11.064
Then we landed at the idea that

00:00:11.064 --> 00:00:13.014
the divergence of V, you know when

00:00:13.014 --> 00:00:15.720
you take the divergence of
this vector valued function,

00:00:15.720 --> 00:00:17.565
it should definitely have something to do

00:00:17.565 --> 00:00:19.863
with the partial derivative of P,

00:00:19.863 --> 00:00:21.224
that X component of the output,

00:00:21.224 --> 00:00:22.791
with respect to X,

00:00:22.791 --> 00:00:24.373
and here I wanna do the opposite,

00:00:24.373 --> 00:00:26.061
and say, okay, what if
we look at functions

00:00:26.061 --> 00:00:30.405
where that P, that
first component is zero,

00:00:30.405 --> 00:00:33.252
but then we have some kind
of positive Q component,

00:00:33.252 --> 00:00:35.792
some kind of positive,
well maybe not positive,

00:00:35.792 --> 00:00:38.171
but some kind of non-zero,

00:00:38.171 --> 00:00:40.605
so positive or negative Y component,

00:00:40.605 --> 00:00:41.703
and what this would mean,

00:00:41.703 --> 00:00:43.961
instead of thinking about
vector as just left and right,

00:00:43.961 --> 00:00:45.790
now we're looking at vectors that

00:00:45.790 --> 00:00:49.081
are purely up or down, kind of up or down.

00:00:49.081 --> 00:00:51.072
So kind of doing the same
thing we did last time,

00:00:51.072 --> 00:00:52.921
if we start thinking about cases where

00:00:52.921 --> 00:00:54.832
the divergence of our function

00:00:54.832 --> 00:00:57.163
at a given point should be positive,

00:00:59.135 --> 00:01:00.587
and an example of that,
you might be saying,

00:01:00.587 --> 00:01:02.905
nothing is happening at the point itself,

00:01:02.905 --> 00:01:06.217
so Q itself would be
zero but then below it,

00:01:06.217 --> 00:01:07.880
things are kind of going away so

00:01:07.880 --> 00:01:09.729
they're pointing down, and above it,

00:01:09.729 --> 00:01:11.151
things are going up.

00:01:11.151 --> 00:01:12.818
So in this case down here,

00:01:12.818 --> 00:01:14.381
Q is a little bit less than zero,

00:01:14.381 --> 00:01:16.779
the Y component of that
vector is less than zero,

00:01:16.779 --> 00:01:19.339
and up here, Q is greater than zero.

00:01:19.339 --> 00:01:20.761
So here we have the idea
that as you're going

00:01:20.761 --> 00:01:22.348
from the bottom up,

00:01:22.348 --> 00:01:24.928
the Y value of your input is increasing

00:01:24.928 --> 00:01:26.798
as you're moving upward in space,

00:01:26.798 --> 00:01:30.068
the value of Q, this Y
component of the output,

00:01:30.068 --> 00:01:31.409
should also be increasing because

00:01:31.409 --> 00:01:34.110
it goes from negative to zero to positive.

00:01:34.110 --> 00:01:35.939
So now you're starting to get this idea

00:01:35.939 --> 00:01:39.497
of partial Q with respect to Y,

00:01:39.497 --> 00:01:41.871
you know, as we change that
Y and move up in space,

00:01:41.871 --> 00:01:44.289
the value of Q should be positive,

00:01:44.289 --> 00:01:45.796
so positive divergence seems to

00:01:45.796 --> 00:01:47.908
correspond to a positive value here.

00:01:47.908 --> 00:01:49.288
And the thinking is actually gonna be

00:01:49.288 --> 00:01:50.447
almost identical to what we did

00:01:50.447 --> 00:01:52.337
in the last video with the X component,

00:01:52.337 --> 00:01:54.041
because you can think
of another circumstance

00:01:54.041 --> 00:01:55.769
where maybe you actually have a vector

00:01:55.769 --> 00:01:57.881
attached to your point
and something's going on,

00:01:57.881 --> 00:01:59.997
and there even is some
convergence towards it,

00:01:59.997 --> 00:02:02.391
where you have some fluid flow

00:02:02.391 --> 00:02:03.550
in towards the point,

00:02:03.550 --> 00:02:05.077
but it's just heavily outweighed by

00:02:05.077 --> 00:02:08.080
even higher divergence, even higher flow

00:02:08.080 --> 00:02:10.136
away from your point above it,

00:02:10.136 --> 00:02:11.537
and again you have this idea of

00:02:11.537 --> 00:02:13.870
Q starts off small, so here's kind of,

00:02:13.870 --> 00:02:16.617
Q starts off small, maybe
it's kind of like near zero,

00:02:16.617 --> 00:02:18.748
and then here Q is something positive

00:02:18.748 --> 00:02:20.271
and then here it's even more positive,

00:02:20.271 --> 00:02:23.889
and sort of making up notation here

00:02:23.889 --> 00:02:25.617
but I want the idea of kind of small

00:02:25.617 --> 00:02:27.487
and then medium-sized and then bigger,

00:02:27.487 --> 00:02:31.549
and once again, the idea
of partial derivative of Q

00:02:31.549 --> 00:02:34.028
with respect to Y being greater than zero

00:02:34.028 --> 00:02:36.487
seems to correspond to
positive divergence,

00:02:36.487 --> 00:02:37.807
and if you want, you can sketch out

00:02:37.807 --> 00:02:39.677
many more circumstances and think about

00:02:39.677 --> 00:02:41.621
what if the vector
started off pointing down,

00:02:41.621 --> 00:02:42.921
what would positive and negative

00:02:42.921 --> 00:02:44.691
and zero divergence all look like,

00:02:44.691 --> 00:02:46.413
but the upshot of it all, pretty much

00:02:46.413 --> 00:02:48.303
for the same reasons I went
through in the last video,

00:02:48.303 --> 00:02:50.702
is this partial derivative
with respect to Y

00:02:50.702 --> 00:02:53.000
corresponds to the divergence.

00:02:53.000 --> 00:02:55.111
And when we combine this with

00:02:55.111 --> 00:02:57.450
our conclusions about the X component,

00:02:57.450 --> 00:03:00.255
that actually is all you need
to know for the divergence.

00:03:00.255 --> 00:03:02.855
So just to write it all out,

00:03:02.855 --> 00:03:05.521
if we have a vector valued
function of X and Y,

00:03:07.383 --> 00:03:09.030
and it's got both of its components,

00:03:09.030 --> 00:03:12.745
we've got P as the X
component of the output,

00:03:12.745 --> 00:03:14.231
that first component of the output,

00:03:14.231 --> 00:03:17.740
and Q, and we're looking
at both of these at once,

00:03:18.890 --> 00:03:21.023
the way that we compute divergence,

00:03:21.023 --> 00:03:25.191
the definition of divergence
of this vector valued function,

00:03:25.191 --> 00:03:28.767
is to say divergence of V

00:03:30.387 --> 00:03:32.854
as a function of X and Y,

00:03:34.917 --> 00:03:38.172
is actually equal to the
partial derivative of P

00:03:38.172 --> 00:03:41.827
with respect to X, plus,

00:03:41.827 --> 00:03:45.836
the partial derivative
of Q with respect to Y.

00:03:47.743 --> 00:03:50.665
And that's it, that is the
formula for divergence,

00:03:50.665 --> 00:03:52.514
and hopefully by now,
this isn't just kind of

00:03:52.514 --> 00:03:54.142
a formula that I'm plopping down for you,

00:03:54.142 --> 00:03:55.845
but it's something that
makes intuitive sense,

00:03:55.845 --> 00:03:58.002
when you see this term, this partial P

00:03:58.002 --> 00:04:00.094
with respect to X, you're thinking about,

00:04:00.094 --> 00:04:01.757
oh yes, yes, because if you have

00:04:01.757 --> 00:04:03.566
flow that's kind of increasing

00:04:03.566 --> 00:04:05.476
as you move in the X direction,

00:04:05.476 --> 00:04:07.527
that's gonna correspond
with movement away,

00:04:07.527 --> 00:04:08.747
and this partial derivative of Q

00:04:08.747 --> 00:04:10.213
with respect to Y term,

00:04:10.213 --> 00:04:11.736
hopefully you're thinking, ah yes,

00:04:11.736 --> 00:04:14.579
as you're increasing the Y component

00:04:14.579 --> 00:04:16.206
of your vector around your point,

00:04:16.206 --> 00:04:19.109
that corresponds with less
flow in than there is out,

00:04:19.109 --> 00:04:21.039
so both of these correspond to that

00:04:21.039 --> 00:04:23.519
idea of divergence that we're going for,

00:04:23.519 --> 00:04:25.126
and if you just add them up,

00:04:25.126 --> 00:04:27.621
this gives you everything
you need to know.

00:04:27.621 --> 00:04:28.901
And one thing that's pretty neat,

00:04:28.901 --> 00:04:30.468
and maybe kind of surprising,

00:04:30.468 --> 00:04:32.761
is that the way we just
came across this formula,

00:04:32.761 --> 00:04:34.000
and started to think about it,

00:04:34.000 --> 00:04:35.100
was in the simplified case,

00:04:35.100 --> 00:04:36.666
where you have, you
know, just pure movement

00:04:36.666 --> 00:04:38.758
in the X direction or pure
movement in the Y direction,

00:04:38.758 --> 00:04:40.179
but in reality, as we know,

00:04:40.179 --> 00:04:42.290
vector fields can look
much more complicated,

00:04:42.290 --> 00:04:43.958
and maybe you have something where

00:04:43.958 --> 00:04:45.626
you know, it's not just
in the X direction,

00:04:45.626 --> 00:04:47.451
and there's lots of things going on

00:04:47.451 --> 00:04:49.018
and you need to account for all of those,

00:04:49.018 --> 00:04:50.681
and evidently, just looking at the change

00:04:50.681 --> 00:04:52.490
in the X component with respect to X,

00:04:52.490 --> 00:04:54.037
or the change in the Y
component of the output

00:04:54.037 --> 00:04:56.310
with respect to the Y
component of the input,

00:04:56.310 --> 00:04:59.970
gives you all the
information you need to know.

00:04:59.970 --> 00:05:01.576
And basically what's going on here is

00:05:01.576 --> 00:05:03.587
that any fluid flow
can just be broken down

00:05:03.587 --> 00:05:05.350
into the X and Y components where

00:05:05.350 --> 00:05:07.119
you're just looking at each vector,

00:05:07.119 --> 00:05:08.908
you know, whatever vector you have,

00:05:08.908 --> 00:05:10.370
it could be broken down into

00:05:10.370 --> 00:05:12.300
it's own X and Y components and if you

00:05:12.300 --> 00:05:13.621
wanna think kind of concretely about

00:05:13.621 --> 00:05:15.918
the fluid flow idea, maybe you'd say that

00:05:15.918 --> 00:05:18.498
for your point, if you're
looking at a point in space,

00:05:18.498 --> 00:05:20.796
you picture a very small box around it,

00:05:20.796 --> 00:05:21.708
and the reason you only need to

00:05:21.708 --> 00:05:23.638
think about X components and Y components

00:05:23.638 --> 00:05:25.488
is that you're only really looking at,

00:05:25.488 --> 00:05:28.088
you know, what's going on
on the left and right side,

00:05:28.088 --> 00:05:29.408
and then you can kind of calculate

00:05:29.408 --> 00:05:31.928
what the divergence
according to fluid flowing

00:05:31.928 --> 00:05:33.289
in through those sides is,

00:05:33.289 --> 00:05:34.977
and then you just look at kind of,

00:05:34.977 --> 00:05:37.517
fluid flowing through
the top or the bottom,

00:05:37.517 --> 00:05:39.507
and if you kinda shrink this box down,

00:05:39.507 --> 00:05:42.838
all you really care about is
those two different directions,

00:05:42.838 --> 00:05:44.809
and anything else, anything that's kind of

00:05:44.809 --> 00:05:47.449
a diagonal into it, is
really just broken down

00:05:47.449 --> 00:05:49.521
into what's the Y component there,

00:05:49.521 --> 00:05:51.537
what's the, you know,
how is it contributing to

00:05:51.537 --> 00:05:53.587
movement up through that
bottom part of the box,

00:05:53.587 --> 00:05:54.827
and then what's the X component,

00:05:54.827 --> 00:05:56.067
how's it contributing to movement

00:05:56.067 --> 00:05:57.830
through that side part of the box.

00:05:57.830 --> 00:05:59.176
But anyway, I mean the upshot here

00:05:59.176 --> 00:06:00.879
is just that the formula for divergence

00:06:00.879 --> 00:06:02.878
only involves these two components.


Divergence example

WEBVTT
Kind: captions
Language: en

00:00:01.018 --> 00:00:03.919
- [Voiceover] So I've got a
vector field here, v of x y.

00:00:03.919 --> 00:00:07.233
Where the first component of
the output is just x times y,

00:00:07.233 --> 00:00:10.057
and the second component is
y squared, minus x squared.

00:00:10.057 --> 00:00:12.766
And the picture of this
vector field is here.

00:00:12.766 --> 00:00:15.006
This is what that vector field looks like.

00:00:15.006 --> 00:00:17.775
And what I'd like to do
is compute and interpret

00:00:17.775 --> 00:00:19.595
the divergence of v.

00:00:19.595 --> 00:00:23.762
So, the divergence of v,
as a function of x and y.

00:00:26.388 --> 00:00:28.815
And in the last couple of
videos I explained that

00:00:28.815 --> 00:00:31.615
the formula for this, and
hopefully it's more than just

00:00:31.615 --> 00:00:33.827
a formula, but something
I have an intuition for,

00:00:33.827 --> 00:00:36.904
is the partial derivative
of p with respect to x.

00:00:36.904 --> 00:00:38.670
By p, I mean that first component.

00:00:38.670 --> 00:00:42.837
So if you're thinking about
this as being p of x y

00:00:44.776 --> 00:00:46.109
and q of x y.

00:00:47.790 --> 00:00:50.582
So I could use any letters
right, and p and q are common.

00:00:50.582 --> 00:00:53.413
But the upshot is it's
the partial derivative of

00:00:53.413 --> 00:00:56.250
the first component with
respect to the first variable.

00:00:56.250 --> 00:01:00.271
Plus, the partial derivative
of that second component,

00:01:00.271 --> 00:01:03.582
with respect to that second variable, y.

00:01:03.582 --> 00:01:06.157
And as we actually plug
this in and start computing

00:01:06.157 --> 00:01:09.338
the partial derivative of p
with respect to x of this guy,

00:01:09.338 --> 00:01:10.571
with respect to x.

00:01:10.571 --> 00:01:12.641
X looks like a variable,
y looks like a constant.

00:01:12.641 --> 00:01:15.558
The derivative is y, that constant.

00:01:16.568 --> 00:01:19.207
And then the partial derivative
of q, that second component,

00:01:19.207 --> 00:01:20.629
with respect to y.

00:01:20.629 --> 00:01:21.957
We look here.

00:01:21.957 --> 00:01:23.779
Y squared looks like a
variable and it's derivative

00:01:23.779 --> 00:01:25.029
is two times y.

00:01:26.587 --> 00:01:28.184
And then x just looks like a constant

00:01:28.184 --> 00:01:29.717
so nothing happens there.

00:01:29.717 --> 00:01:32.433
So in the whole, the divergence
evidently just depends on

00:01:32.433 --> 00:01:33.791
the y value.

00:01:33.791 --> 00:01:35.624
It's three times y.

00:01:35.624 --> 00:01:37.708
So what that should mean is
if we look at, for example,

00:01:37.708 --> 00:01:40.300
let's say we look our points
for y equals zero, we'd expect

00:01:40.300 --> 00:01:42.203
the divergence to be zero.

00:01:42.203 --> 00:01:45.425
The fluid neither goes towards
nor away from each point.

00:01:45.425 --> 00:01:48.661
So y equals zero corresponds
with this x axis of points.

00:01:48.661 --> 00:01:50.946
So to give it a point here,
evidently it's the case

00:01:50.946 --> 00:01:54.786
that the fluid kind of flowing
in from above is bounced out

00:01:54.786 --> 00:01:57.139
by how much fluid flows away from it here

00:01:57.139 --> 00:01:58.092
and wherever you look.

00:01:58.092 --> 00:02:00.354
I mean, here its only
flowing in by a little bit,

00:02:00.354 --> 00:02:02.947
and I guess it's flowing
out just by that same amount

00:02:02.947 --> 00:02:04.782
and that all cancels out.

00:02:04.782 --> 00:02:08.949
Whereas, let's say we take
a look at y equals three.

00:02:10.524 --> 00:02:13.123
So in this case the
divergence should equal nine.

00:02:13.123 --> 00:02:15.956
So we'd expect there to be
positive divergence when y

00:02:15.956 --> 00:02:17.228
is positive.

00:02:17.228 --> 00:02:20.801
So if we go up, and y is
equal to one, two, three.

00:02:20.801 --> 00:02:24.735
And if we look at a point
around here, I'm gonna

00:02:24.735 --> 00:02:26.710
kinda consider the region around it.

00:02:26.710 --> 00:02:29.078
You can kinda see how the
vectors leaving it seem

00:02:29.078 --> 00:02:30.119
to be bigger.

00:02:30.119 --> 00:02:34.066
So the fluid flowing out of
this region is pretty rapid.

00:02:34.066 --> 00:02:36.855
Whereas the fluid flowing
into it is less rapid.

00:02:36.855 --> 00:02:39.261
So on the whole, in a region
around this point, the fluid

00:02:39.261 --> 00:02:40.943
I guess is going away.

00:02:40.943 --> 00:02:43.043
And you look anywhere where y is positive

00:02:43.043 --> 00:02:45.739
and if you kind of look
around here, the same is true,

00:02:45.739 --> 00:02:48.424
where fluid does flow into it, it seems.

00:02:48.424 --> 00:02:51.965
But the vectors kind of going
out of this region are larger.

00:02:51.965 --> 00:02:54.399
So you'd expect on the whole
for things to diverge away

00:02:54.399 --> 00:02:55.732
from that point.

00:02:57.266 --> 00:02:59.927
In contrast, if you look at
something where y is negative,

00:02:59.927 --> 00:03:03.199
let's say it was y is equal
to negative four, it doesn't

00:03:03.199 --> 00:03:04.032
have to be three there.

00:03:04.032 --> 00:03:06.440
So there would be a
divergence of negative 12.

00:03:06.440 --> 00:03:08.832
So you'd expect things to
definitely converging towards

00:03:08.832 --> 00:03:10.849
your specific points.

00:03:10.849 --> 00:03:13.972
So you go down to, I guess I
said y equals negative four.

00:03:13.972 --> 00:03:17.328
But really, I'm thinking of
anything where y is negative.

00:03:17.328 --> 00:03:19.770
Let's say we take a
look at this point here.

00:03:19.770 --> 00:03:22.111
Fluid flowing into it
seems to be according

00:03:22.111 --> 00:03:22.981
to large vectors.

00:03:22.981 --> 00:03:24.984
So it's flowing into
it pretty quickly here.

00:03:24.984 --> 00:03:27.607
But when it's flowing out
of it, it's less large.

00:03:27.607 --> 00:03:30.925
It's flowing out of it in a
kind of a lackadaisical way.

00:03:30.925 --> 00:03:34.125
So, it kinda makes sense,
just looking at the picture

00:03:34.125 --> 00:03:38.055
the divergence tends to be
negative when y is negative.

00:03:38.055 --> 00:03:40.037
And what's surprising, What
I wouldn't have been able

00:03:40.037 --> 00:03:41.805
to tell just looking
at the picture, is that

00:03:41.805 --> 00:03:44.587
the divergence only
depends on the y value.

00:03:44.587 --> 00:03:48.113
But once you compute everything,
it's only dependent on

00:03:48.113 --> 00:03:49.363
the y value here.

00:03:49.363 --> 00:03:52.099
And that as you go kind
of left and right on

00:03:52.099 --> 00:03:53.599
the diagram there.

00:03:54.786 --> 00:03:57.397
As we look left and right, the value of

00:03:57.397 --> 00:03:58.585
the divergence doesn't change.

00:03:58.585 --> 00:03:59.725
That's kind of surprising.

00:03:59.725 --> 00:04:01.038
It makes a little bit of sense.

00:04:01.038 --> 00:04:03.748
You don't see any notable
reason that the divergence here

00:04:03.748 --> 00:04:05.179
should be any different than here.

00:04:05.179 --> 00:04:09.346
But, I wouldn't have known that
they were exactly the same.


Node voltage method (steps 1 to 4)

WEBVTT
Kind: captions
Language: en

00:00:00.762 --> 00:00:01.720
- [Voiceover] We're going to talk about

00:00:01.720 --> 00:00:03.984
a really powerful way to analyze circuits

00:00:03.984 --> 00:00:07.262
called the Node Voltage Method.

00:00:07.262 --> 00:00:09.707
Before we start talking
about what this method is,

00:00:09.707 --> 00:00:13.874
we're going to talk about a
new term called a node voltage.

00:00:15.353 --> 00:00:18.270
So far, we already have the idea of

00:00:20.053 --> 00:00:24.220
an element has a voltage
across it, and we refer to that

00:00:25.337 --> 00:00:28.870
as an element voltage, or
if it's part of a circuit

00:00:28.870 --> 00:00:30.255
and it's a branch of a circuit,

00:00:30.255 --> 00:00:32.922
it'd be called a branch voltage.

00:00:40.636 --> 00:00:41.994
That's a voltage that's associated

00:00:41.994 --> 00:00:44.486
with a particular element.

00:00:44.486 --> 00:00:48.653
So now we have the idea of
something called a node voltage.

00:00:53.634 --> 00:00:55.907
This is still a voltage,
it's not anything strange,

00:00:55.907 --> 00:00:57.871
but if we go over to our circuit here,

00:00:57.871 --> 00:00:58.840
and we label the nodes.

00:00:58.840 --> 00:01:01.664
Let's start labeling the nodes,
we'll call this node here

00:01:01.664 --> 00:01:05.003
where this junction between
this resistor and this source,

00:01:05.003 --> 00:01:06.396
we'll call this node one.

00:01:06.396 --> 00:01:10.041
This is the junction between
these two components here.

00:01:10.041 --> 00:01:13.728
There's another node that's
these two resistors connected

00:01:13.728 --> 00:01:16.413
to this current source,
and that's a single

00:01:16.413 --> 00:01:20.600
distributed node, so
we'll call that node two.

00:01:20.600 --> 00:01:23.771
And down here, these three
components are connected together

00:01:23.771 --> 00:01:26.854
in a junction, and that's node three.

00:01:27.717 --> 00:01:31.104
To define a node voltage,
the first idea we need

00:01:31.104 --> 00:01:35.271
is to define a reference node,
the idea of a reference node.

00:01:44.304 --> 00:01:46.699
A good choice for the
reference node is usually one

00:01:46.699 --> 00:01:50.980
that's connected to the
terminals of the power sources,

00:01:50.980 --> 00:01:55.723
or it's the node that's
connected to a lot of branches,

00:01:55.723 --> 00:01:57.913
a lot of elements, and node three here

00:01:57.913 --> 00:02:00.181
is a good choice for a reference node.

00:02:00.181 --> 00:02:04.662
The way we mark that is with
a symbol that looks like this,

00:02:04.662 --> 00:02:09.104
a ground symbol, that's
called ground in this circuit.

00:02:09.104 --> 00:02:13.082
There's other kinds of ways
to indicate a reference node.

00:02:13.082 --> 00:02:14.641
That's a common way.

00:02:14.641 --> 00:02:18.503
You can draw one that
looks like the ground,

00:02:18.503 --> 00:02:20.171
connected to the ground.

00:02:20.171 --> 00:02:23.758
Sometimes you'll see it
with just an upside-down T,

00:02:23.758 --> 00:02:27.675
like that, that's another
way to draw a ground.

00:02:28.586 --> 00:02:32.014
This symbol on a schematic
indicates the reference node.

00:02:32.014 --> 00:02:36.445
We've picked a reference node
to be node three, down here.

00:02:36.445 --> 00:02:40.195
So, a node voltage is
measured between a node

00:02:41.812 --> 00:02:43.747
and the reference node.

00:02:43.747 --> 00:02:46.914
In our case we have this voltage here,

00:02:48.362 --> 00:02:52.529
is the node voltage on
node one, we'll call it V1.

00:02:56.605 --> 00:03:00.188
This voltage here is
going to be called V2.

00:03:02.179 --> 00:03:04.062
And in particular, these
voltages are measured

00:03:04.062 --> 00:03:08.229
with respect to node three,
so there's the minus and plus

00:03:09.372 --> 00:03:10.955
and minus and plus.

00:03:11.861 --> 00:03:13.767
We're going to use these node voltages

00:03:13.767 --> 00:03:16.017
in the Node Voltage Method.

00:03:17.063 --> 00:03:18.080
First, what I want to do,

00:03:18.080 --> 00:03:21.714
I want to label my complements here.

00:03:21.714 --> 00:03:25.881
We're going to call this
Vs, and make it 15 volts.

00:03:26.983 --> 00:03:30.083
This resistor's going to
be R1, and we'll give it

00:03:30.083 --> 00:03:31.583
a value of 4kohms.

00:03:35.094 --> 00:03:39.261
We'll call this R2, and we'll
give it a value of 2kohms.

00:03:42.853 --> 00:03:45.827
This is the same circuit
that we analyzed when we did

00:03:45.827 --> 00:03:50.428
application of the fundamental
laws in another video.

00:03:50.428 --> 00:03:54.511
Oh, and the last guy here,
Is, current source Is,

00:03:55.406 --> 00:03:58.406
and we'll make that one 3 milliamps.

00:04:01.623 --> 00:04:03.058
We've analyzed this circuit before.

00:04:03.058 --> 00:04:07.907
We used Kirchoff's Laws,
KVL and KCL, to figure out

00:04:07.907 --> 00:04:10.589
what the voltages and
currents were in this circuit.

00:04:10.589 --> 00:04:11.940
We're going to do the same analysis,

00:04:11.940 --> 00:04:13.890
but this time we're
going to use what we call

00:04:13.890 --> 00:04:16.073
the Node Voltage Method.

00:04:16.073 --> 00:04:18.190
It's basically the same application

00:04:18.190 --> 00:04:21.732
of the fundamental laws, we
use Ohm's Law, Kirchoff's Laws,

00:04:21.732 --> 00:04:24.366
but it's in a really
clever, organized way,

00:04:24.366 --> 00:04:26.773
that is really efficient.

00:04:26.773 --> 00:04:29.104
Whoever thought this up was pretty bright

00:04:29.104 --> 00:04:31.815
and I'm really glad
that they wrote it down

00:04:31.815 --> 00:04:33.615
and shared it with us.

00:04:33.615 --> 00:04:35.781
What I want to do first is just write down

00:04:35.781 --> 00:04:38.306
what are the steps of this method?

00:04:38.306 --> 00:04:40.458
It's not a theory, it's
a method, so it basically

00:04:40.458 --> 00:04:42.453
a sequence of steps that you go through

00:04:42.453 --> 00:04:43.890
to analyze the circuit.

00:04:43.890 --> 00:04:45.823
I'll write the list right here.

00:04:45.823 --> 00:04:48.823
First step is pick a reference node.

00:04:51.659 --> 00:04:53.326
We already did that.

00:04:58.469 --> 00:05:02.219
The second step is to
name the node voltages.

00:05:10.415 --> 00:05:13.742
We already did that,
we named our nodes V1,

00:05:13.742 --> 00:05:16.363
that node there is V1 and
that node there is V2,

00:05:16.363 --> 00:05:19.207
with respect to the reference node,

00:05:19.207 --> 00:05:21.424
which is down there at node three.

00:05:21.424 --> 00:05:23.290
Whenever you talk about node voltages,

00:05:23.290 --> 00:05:25.014
there's always an assumption
that one of the nodes

00:05:25.014 --> 00:05:26.681
is a reference node.

00:05:32.031 --> 00:05:35.531
The third step is to solve the easy nodes.

00:05:42.516 --> 00:05:44.805
I'll show you what that means in a second.

00:05:44.805 --> 00:05:47.472
The fourth step is to write KCL,

00:05:51.804 --> 00:05:54.554
Kirchoff's Current Law equations.

00:06:02.237 --> 00:06:05.654
The fifth step is to solve the equations.

00:06:09.723 --> 00:06:13.053
That's the Node Voltage Method,

00:06:13.053 --> 00:06:14.307
and we're going to go
through the rest of this,

00:06:14.307 --> 00:06:16.890
we've done the first two steps.

00:06:18.403 --> 00:06:20.928
What does it mean to solve the easy nodes?

00:06:20.928 --> 00:06:22.962
The easy nodes are the
ones that are connected

00:06:22.962 --> 00:06:25.992
directly to a source that
goes to the reference node.

00:06:25.992 --> 00:06:27.778
That's an example of an easy node.

00:06:27.778 --> 00:06:29.910
So V1 is an easy node.

00:06:29.910 --> 00:06:32.509
So let's solve for V1.

00:06:32.509 --> 00:06:35.842
By inspection, I can say V1 is 15 volts.

00:06:40.052 --> 00:06:41.650
That's Step Three.

00:06:41.650 --> 00:06:43.023
The other node's not easy,

00:06:43.023 --> 00:06:44.786
the other node has lots of components

00:06:44.786 --> 00:06:47.853
and something interesting's
going on over here.

00:06:47.853 --> 00:06:50.075
So this was step three.

00:06:50.075 --> 00:06:51.461
Let's label the steps.

00:06:51.461 --> 00:06:53.128
Here's the Step One.

00:06:54.175 --> 00:06:55.508
Here's Step Two.

00:06:57.560 --> 00:06:59.393
And here's Step Three.

00:07:03.534 --> 00:07:06.043
Now we're ready to go to Step Four,

00:07:06.043 --> 00:07:08.376
let me move up a little bit.

00:07:12.710 --> 00:07:17.001
Step Four is write the
Kirchoff's Current Law equations

00:07:17.001 --> 00:07:18.427
directly from the circuit.

00:07:18.427 --> 00:07:22.018
We're going to do this in a special way,

00:07:22.018 --> 00:07:25.011
We're going to perform at
this node here, at node two.

00:07:25.011 --> 00:07:28.855
We're going to write the
current law for this.

00:07:28.855 --> 00:07:31.654
That means we got to
identify the currents.

00:07:31.654 --> 00:07:34.913
There's a current, we'll
call that a current,

00:07:34.913 --> 00:07:37.337
and that's a current.

00:07:37.337 --> 00:07:40.374
Let me give some names to these
currents just to be clear.

00:07:40.374 --> 00:07:44.071
We'll call this one I1
because it goes through R1.

00:07:44.071 --> 00:07:48.238
We'll call this one here, I2
because it goes through R2.

00:07:50.887 --> 00:07:52.804
This one is already Is.

00:07:55.871 --> 00:07:59.051
Now let's write Kirchoff's
Current Law just in terms of I,

00:07:59.051 --> 00:08:02.178
and we'll say all the
currents flowing into the node

00:08:02.178 --> 00:08:06.002
add up to zero, so these
two have arrows going out,

00:08:06.002 --> 00:08:08.641
so they're going to get negative signs

00:08:08.641 --> 00:08:10.631
when we write Kirchoff's Current Law.

00:08:10.631 --> 00:08:12.714
Let's do that right here.

00:08:16.458 --> 00:08:17.708
And we write I1

00:08:20.216 --> 00:08:21.049
minus I2

00:08:23.015 --> 00:08:23.848
minus Is

00:08:25.865 --> 00:08:27.115
equals to zero.

00:08:31.283 --> 00:08:34.616
So right now we're working on Step Four.

00:08:38.423 --> 00:08:41.537
This is the essence of
the Node Voltage Method.

00:08:41.537 --> 00:08:42.638
This is where we do something new

00:08:42.638 --> 00:08:44.138
that we haven't done before.

00:08:44.138 --> 00:08:45.473
We're going to write these currents

00:08:45.473 --> 00:08:48.607
in terms of the node voltages.

00:08:48.607 --> 00:08:52.761
So we can write I1, I1 is
current flowing this way

00:08:52.761 --> 00:08:53.977
through this current.

00:08:53.977 --> 00:08:54.977
I1 equals V1

00:08:58.418 --> 00:08:59.251
minus V2

00:09:00.879 --> 00:09:01.712
over R1.

00:09:06.009 --> 00:09:08.824
That's the current flowing in resistor R1,

00:09:08.824 --> 00:09:10.991
in terms of node voltages.

00:09:13.385 --> 00:09:16.525
The current flowing down through I2,

00:09:16.525 --> 00:09:20.660
now we have to subtract I2,
so we just apply Ohm's Law

00:09:20.660 --> 00:09:23.768
directly, which means
that the current in I2

00:09:23.768 --> 00:09:26.185
is equal to V2 divided by R2.

00:09:30.701 --> 00:09:33.451
The last current is Is, minus Is.

00:09:35.849 --> 00:09:38.767
We'll write that in
terms of Is, like that,

00:09:38.767 --> 00:09:40.517
and that equals zero.

00:09:41.891 --> 00:09:44.709
This means we have now
completed Step Four.

00:09:44.709 --> 00:09:48.876
That is KCL written using the
terminology of node voltages.

00:09:51.400 --> 00:09:55.150
We could check off that
we've done Step Four.


Ionic bonds and Coulombs law

WEBVTT
Kind: captions
Language: en

00:00:00.325 --> 00:00:01.611
- [Voiceover] Ionic bonds are the bonds

00:00:01.611 --> 00:00:04.026
that hold together ionic compounds.

00:00:04.026 --> 00:00:07.254
So basically it's what holds
together cations and anions.

00:00:07.254 --> 00:00:09.297
An example of a compound that's

00:00:09.297 --> 00:00:10.992
held together with ionic bonds

00:00:10.992 --> 00:00:15.032
is sodium chloride, also
known as table salt.

00:00:15.032 --> 00:00:16.797
So here we have a close-up picture

00:00:16.797 --> 00:00:19.816
of some really nice
crystals of sodium chloride.

00:00:19.816 --> 00:00:21.952
And this is something that
you could try at home.

00:00:21.952 --> 00:00:25.156
You can take any table
salt that you might have,

00:00:25.156 --> 00:00:27.455
dissolve it up in some water,

00:00:27.455 --> 00:00:29.731
and then let that water slowly evaporate.

00:00:29.731 --> 00:00:31.170
And if you're lucky, you might get some

00:00:31.170 --> 00:00:34.073
beautiful symmetric crystals like these.

00:00:34.073 --> 00:00:35.419
For me personally at least,

00:00:35.419 --> 00:00:37.486
growing crystals that look beautiful

00:00:37.486 --> 00:00:40.737
is one of the most fun
things about chemistry.

00:00:40.737 --> 00:00:42.339
You can see when we look at the

00:00:42.339 --> 00:00:44.754
close-up shapes of these crystals

00:00:44.754 --> 00:00:47.331
that they have some
very beautiful symmetry.

00:00:47.331 --> 00:00:49.351
That symmetry tells us a little bit

00:00:49.351 --> 00:00:51.743
about the structure of these compounds

00:00:51.743 --> 00:00:53.740
on a molecular level.

00:00:53.740 --> 00:00:55.644
If we zoom in on these crystals,

00:00:55.644 --> 00:00:59.429
we can imagine, actually
we don't have to imagine,

00:00:59.429 --> 00:01:02.285
you can look at these with
different kinds of instruments

00:01:02.285 --> 00:01:03.678
like x-ray crystallography,

00:01:03.678 --> 00:01:05.977
and you can look at the crystal lattice

00:01:05.977 --> 00:01:07.708
and get information about

00:01:07.708 --> 00:01:11.875
how the different ions are
arranged in the solids.

00:01:12.757 --> 00:01:14.638
So the way that the ions are arranged

00:01:14.638 --> 00:01:16.240
determines a lot of things

00:01:16.240 --> 00:01:18.469
about the properties of these compounds.

00:01:18.469 --> 00:01:22.277
So these ionic bonds and
how the ions are arranged

00:01:22.277 --> 00:01:26.736
tell us a lot about the
solubility of the compound.

00:01:26.736 --> 00:01:27.653
Solubility.

00:01:29.754 --> 00:01:33.921
And other properties like
melting or boiling points.

00:01:35.327 --> 00:01:37.510
And it even can be related back to things

00:01:37.510 --> 00:01:41.010
like how hard a particular ionic solid is.

00:01:43.106 --> 00:01:46.147
So the ionic bonds here,
in the sodium chloride,

00:01:46.147 --> 00:01:47.935
are the ones that hold together

00:01:47.935 --> 00:01:50.397
our sodium ions and our chloride ions.

00:01:50.397 --> 00:01:52.974
So our sodium plus and our chloride minus.

00:01:52.974 --> 00:01:54.599
And the strength of an ionic bond

00:01:54.599 --> 00:01:57.766
is related to the electrostatic force.

00:02:00.729 --> 00:02:03.812
The electrostatic force between them.

00:02:04.697 --> 00:02:08.639
And I'm going to abbreviate
the electrostatic force

00:02:08.639 --> 00:02:10.056
as F subscript e.

00:02:11.268 --> 00:02:16.098
So this is the force
between two charged species.

00:02:16.098 --> 00:02:19.098
And it's equal to some constant k...

00:02:25.804 --> 00:02:28.730
Times the two charges that are interacting

00:02:28.730 --> 00:02:32.897
divided by the distance between
the two charges squared.

00:02:34.256 --> 00:02:37.173
So here, q1 and q2 are the charges,

00:02:38.134 --> 00:02:42.569
and in the case of sodium
chloride for example,

00:02:42.569 --> 00:02:44.319
q1 and q2 would be...

00:02:46.145 --> 00:02:49.645
q1 might be one plus, from our sodium ion,

00:02:51.300 --> 00:02:55.383
and q2 might be one minus,
from our chloride ion.

00:02:56.942 --> 00:02:58.753
And we could also just switch those two,

00:02:58.753 --> 00:03:01.888
we could say chloride
is q1 and sodium is q2,

00:03:01.888 --> 00:03:06.067
and that wouldn't change what
we get from this equation.

00:03:06.067 --> 00:03:09.318
And then r2 here, is the
distance between the ions,

00:03:09.318 --> 00:03:11.965
and we usually approximate it as saying

00:03:11.965 --> 00:03:14.798
it's the sum of the ionic radii...

00:03:18.049 --> 00:03:20.417
for the two ions we're looking at.

00:03:20.417 --> 00:03:22.670
So we can use Coulomb's law here

00:03:22.670 --> 00:03:24.736
to explain some properties
that are related

00:03:24.736 --> 00:03:26.872
to the strengths of ionic bonds.

00:03:26.872 --> 00:03:29.868
And so the example we're
gonna go through today

00:03:29.868 --> 00:03:32.956
is going to be that of melting point.

00:03:32.956 --> 00:03:35.487
So we're gonna look at
some melting point trends

00:03:35.487 --> 00:03:37.461
and try to relate them to the

00:03:37.461 --> 00:03:40.544
different variables in Coulomb's law.

00:03:41.431 --> 00:03:43.591
So the first thing we'll look at,

00:03:43.591 --> 00:03:46.517
the first two compounds we'll compare

00:03:46.517 --> 00:03:48.350
are sodium fluoride...

00:03:50.441 --> 00:03:52.108
and magnesium oxide.

00:03:55.478 --> 00:03:59.645
Sodium fluoride has a melting
point of 993 degrees Celsius,

00:04:01.749 --> 00:04:04.791
and magnesium oxide has a melting point

00:04:04.791 --> 00:04:06.791
of 2852 degrees Celsius.

00:04:11.942 --> 00:04:14.775
The other information we know
about these two compounds,

00:04:14.775 --> 00:04:16.819
if you look up the ionic radii,

00:04:16.819 --> 00:04:18.723
it turns out that sodium fluoride,

00:04:18.723 --> 00:04:20.513
the distance between the ions

00:04:20.513 --> 00:04:23.529
is about the same as magnesium oxide.

00:04:23.529 --> 00:04:26.525
They're not exactly the same,
but they're pretty close,

00:04:26.525 --> 00:04:28.498
so if we were to say that r is

00:04:28.498 --> 00:04:30.565
approximately the same for these two,

00:04:30.565 --> 00:04:32.422
then we can explain the difference

00:04:32.422 --> 00:04:35.186
in melting points using the charges.

00:04:35.186 --> 00:04:37.531
Since melting point is a measure of,

00:04:37.531 --> 00:04:39.644
basically how much energy do you need

00:04:39.644 --> 00:04:42.639
to add to these compounds
to break apart your ions,

00:04:42.639 --> 00:04:46.238
we would expect melting point to go up,

00:04:46.238 --> 00:04:48.655
to increase, as Fe increases.

00:04:52.252 --> 00:04:54.481
As the force between the ions increases,

00:04:54.481 --> 00:04:57.268
we would expect to have to add more energy

00:04:57.268 --> 00:04:59.938
to break those ions apart.

00:04:59.938 --> 00:05:02.423
And we can see that in our first example.

00:05:02.423 --> 00:05:05.279
Magnesium oxide, if we look
at the charges on the ions,

00:05:05.279 --> 00:05:09.412
magnesium is two plus,
and oxide is two minus.

00:05:09.412 --> 00:05:12.523
In sodium fluoride, sodium is one plus,

00:05:12.523 --> 00:05:14.776
and fluoride is one minus.

00:05:14.776 --> 00:05:18.632
So we would expect, assuming
that r is about the same,

00:05:18.632 --> 00:05:22.369
this q1 times q2 is four times bigger

00:05:22.369 --> 00:05:25.869
in magnesium oxide versus sodium fluoride.

00:05:27.441 --> 00:05:30.858
So q1 and q2, the product of q1 and q2...

00:05:33.073 --> 00:05:35.573
is higher for magnesium oxide,

00:05:36.626 --> 00:05:38.321
and that's why we would expect

00:05:38.321 --> 00:05:40.916
the melting point to be higher.

00:05:40.916 --> 00:05:45.083
We can also look at sodium
chloride versus sodium fluoride.

00:05:52.859 --> 00:05:55.132
And in this case, let's look at,

00:05:55.132 --> 00:05:57.361
well, I don't know, maybe
this is kind of artificial,

00:05:57.361 --> 00:05:59.033
the boiling point, the melting point,

00:05:59.033 --> 00:06:02.383
sorry, the melting
point of sodium chloride

00:06:02.383 --> 00:06:04.466
is 801 degrees Celsius...

00:06:08.016 --> 00:06:11.290
and the melting point
of sodium fluoride is,

00:06:11.290 --> 00:06:14.623
like we said early, 993 degrees Celsius.

00:06:17.118 --> 00:06:20.857
And so this time, the charges
are the same on our ions,

00:06:20.857 --> 00:06:24.874
our q1 and q2 is one plus for
the sodium in both compounds,

00:06:24.874 --> 00:06:29.123
and one minus for the
chloride and the fluoride.

00:06:29.123 --> 00:06:32.490
So q1 times q2 didn't change
for these two compounds,

00:06:32.490 --> 00:06:36.657
but since we changed the anion
from fluoride to chloride,

00:06:37.751 --> 00:06:39.418
we increased r here,

00:06:41.104 --> 00:06:43.821
and increasing r in the denominator makes

00:06:43.821 --> 00:06:47.211
the electrostatic force goes down.

00:06:47.211 --> 00:06:50.810
Another way we could put it
is that since r decreases,

00:06:50.810 --> 00:06:54.804
as we go from sodium
chloride to sodium fluoride,

00:06:54.804 --> 00:06:56.971
the melting point goes up.

00:06:58.241 --> 00:06:59.982
So in each of these pairs,

00:06:59.982 --> 00:07:03.861
the compound that has
the higher melting point

00:07:03.861 --> 00:07:05.778
is the one that also...

00:07:08.485 --> 00:07:11.612
has the higher electrostatic forces,

00:07:11.612 --> 00:07:14.399
and that's either because
the charges are higher,

00:07:14.399 --> 00:07:15.838
q1 and q2 are higher,

00:07:15.838 --> 00:07:19.275
or because the distance
between the ions went down.

00:07:19.275 --> 00:07:22.201
So these are some examples
for how we can relate

00:07:22.201 --> 00:07:24.174
the properties of ionic compounds

00:07:24.174 --> 00:07:27.750
to the electrostatic
force using Coulomb's law

00:07:27.750 --> 00:07:30.500
between the cation and the anion.


Calculating internal energy and work example

WEBVTT
Kind: captions
Language: en

00:00:00.001 --> 00:00:00.784
- [Voiceover] In this video

00:00:00.784 --> 00:00:02.463
we're gonna do an example problem,

00:00:02.463 --> 00:00:04.645
where we calculate in internal energy

00:00:04.645 --> 00:00:07.693
and also calculate pressure volume work.

00:00:08.707 --> 00:00:11.861
So, we know the external pressure

00:00:11.861 --> 00:00:15.464
is 1.01 times ten to the 5th pascal,

00:00:15.464 --> 00:00:18.694
and our system is some balloon,

00:00:18.694 --> 00:00:20.915
let's say it's a balloon of argon gas,

00:00:21.849 --> 00:00:26.362
and initially our gas has
a volume of 2.3 liters,

00:00:27.435 --> 00:00:30.029
and then it transfers,

00:00:30.029 --> 00:00:35.029
the gas transfers 485 joules

00:00:35.481 --> 00:00:38.597
of energy as heat to the surroundings.

00:00:44.489 --> 00:00:45.453
Once it does that,

00:00:45.453 --> 00:00:47.407
the final volume of our system,

00:00:47.407 --> 00:00:50.689
is 2.05 liters.

00:00:50.689 --> 00:00:51.908
And we're assuming here,

00:00:51.908 --> 00:00:54.809
that the moles of gas didn't change.

00:00:55.949 --> 00:00:58.767
The question we're gonna answer is,

00:00:58.767 --> 00:01:03.238
for this process, what is delta U?

00:01:03.238 --> 00:01:06.355
So, what the change in
internal energy for our system?

00:01:07.221 --> 00:01:10.808
We can use the first law of
thermodynamics to answer this.

00:01:10.808 --> 00:01:14.140
The first law tells us that
the change in internal energy,

00:01:14.140 --> 00:01:18.621
delta U, is equal to the work done,

00:01:20.101 --> 00:01:21.849
plus the heat transfer.

00:01:21.849 --> 00:01:23.476
Before we plug any numbers in here,

00:01:23.476 --> 00:01:25.849
the first thing I wanna do is make sure

00:01:25.849 --> 00:01:28.816
I have a good idea of what
signs everything should be.

00:01:28.816 --> 00:01:29.989
I think that's one of the trickiest things

00:01:29.989 --> 00:01:31.816
in these kind of problems.

00:01:31.816 --> 00:01:35.027
So here, since our
system transferred energy

00:01:35.027 --> 00:01:37.291
to the surroundings and
not the other way around,

00:01:37.291 --> 00:01:39.590
Q should be negative,

00:01:39.590 --> 00:01:42.029
because when your system transfers energy

00:01:42.029 --> 00:01:43.189
to the surroundings,

00:01:43.189 --> 00:01:45.919
then it's internal energy should go down.

00:01:45.919 --> 00:01:47.800
Work on the other hand,

00:01:47.800 --> 00:01:51.255
since V two is less than V one,

00:01:51.255 --> 00:01:52.988
the volume of our system went down,

00:01:52.988 --> 00:01:54.428
which means the surroundings

00:01:54.428 --> 00:01:56.898
had to do work on the gas

00:01:56.898 --> 00:01:58.729
to get the volume to decrease.

00:01:58.729 --> 00:02:00.410
We would expect if the surroundings

00:02:00.410 --> 00:02:02.151
did work on our system,

00:02:02.151 --> 00:02:04.056
that would increase the internal energy.

00:02:04.056 --> 00:02:07.631
So that means the work
done here is positive.

00:02:07.631 --> 00:02:09.411
We can also calculate work

00:02:09.411 --> 00:02:11.385
because we know the external pressure,

00:02:11.385 --> 00:02:13.126
we know it's constant,

00:02:13.126 --> 00:02:15.915
and work can be calculated

00:02:15.915 --> 00:02:19.635
as the external pressure
times the change in volume,

00:02:19.635 --> 00:02:20.890
and we know both of those things,

00:02:20.890 --> 00:02:22.277
we know the external pressure

00:02:22.277 --> 00:02:25.545
and we know the initial and final volumes.

00:02:25.545 --> 00:02:27.252
So if we start plugging that in,

00:02:28.602 --> 00:02:30.720
we get that delta U is equal

00:02:30.720 --> 00:02:34.488
to negative 485 joules,

00:02:35.597 --> 00:02:38.511
so that's our heat, we know
it should have a negative sign

00:02:38.511 --> 00:02:41.576
because the heat was
transferred to the surroundings.

00:02:41.576 --> 00:02:44.141
So negative 485 joules,

00:02:44.141 --> 00:02:47.282
minus, we should have
a negative sign there,

00:02:48.379 --> 00:02:52.230
minus the external pressure 1.01,

00:02:52.230 --> 00:02:55.271
times ten to the 5th pascals,

00:02:56.959 --> 00:02:59.699
so that's our external pressure
times our change in volume.

00:02:59.699 --> 00:03:01.139
So that's our final pressure,

00:03:01.139 --> 00:03:05.039
2.05 liters, minus our initial pressure,

00:03:05.039 --> 00:03:07.972
which is 2.30 liters.

00:03:08.940 --> 00:03:10.356
We could at this point be like," okay,

00:03:10.356 --> 00:03:11.518
"we figured it all out,

00:03:11.518 --> 00:03:12.900
"we just have to stick
all of these numbers

00:03:12.900 --> 00:03:15.137
"in a calculator and we're done,"

00:03:15.137 --> 00:03:19.384
and that's probably what my
first instinct would often be,

00:03:19.384 --> 00:03:21.644
but there's one more
thing that we should check

00:03:21.644 --> 00:03:23.340
before we actually plug in the numbers

00:03:23.340 --> 00:03:25.487
and have a party.

00:03:25.487 --> 00:03:26.937
And that's our units,

00:03:26.937 --> 00:03:31.035
we have our heat, in terms of joules,

00:03:31.035 --> 00:03:33.555
we probably want our
change in internal energy

00:03:33.555 --> 00:03:35.738
in terms of joules too.

00:03:35.738 --> 00:03:36.922
On this other side,

00:03:36.922 --> 00:03:38.582
we're calculating our work here,

00:03:38.582 --> 00:03:43.435
and we have pascals, so pascals times,

00:03:43.435 --> 00:03:46.141
so we have joules, and joules,

00:03:46.141 --> 00:03:48.562
we have pascals times liters,

00:03:48.562 --> 00:03:49.860
so then the question is, okay,

00:03:49.860 --> 00:03:54.860
we're doing joules minus
pascals, times liters,

00:03:56.125 --> 00:03:59.063
we need to make sure that
whatever we calculate here,

00:03:59.063 --> 00:04:02.080
in terms of work, also
has units of joules.

00:04:02.080 --> 00:04:04.937
Otherwise, we will be, we'll
be subtracting two things

00:04:04.937 --> 00:04:06.364
that don't have the same unit,

00:04:06.364 --> 00:04:08.246
and that's bad.
(giggles)

00:04:08.246 --> 00:04:11.021
We'll have to do some sort
of unit conversion first.

00:04:11.021 --> 00:04:14.340
So, let's just double check
that pascals times liters,

00:04:14.340 --> 00:04:16.338
will give us joules.

00:04:16.338 --> 00:04:17.742
So the way I did this,

00:04:17.742 --> 00:04:21.922
is by converting everything
to the same units.

00:04:21.922 --> 00:04:22.978
So, if you take joules,

00:04:22.978 --> 00:04:24.337
which is already SI units,

00:04:24.337 --> 00:04:26.264
we can actually simplify it more,

00:04:26.264 --> 00:04:28.795
in terms of other SI units.

00:04:28.795 --> 00:04:30.794
So, a joule is equal

00:04:30.794 --> 00:04:35.774
to one kilogram meters squared

00:04:35.774 --> 00:04:36.929
per second squared.

00:04:37.825 --> 00:04:39.984
So, joules is equal to, one joule is equal

00:04:39.984 --> 00:04:42.847
to one kilogram meters
squared over seconds squared.

00:04:44.025 --> 00:04:47.542
One pascal, pascals are also in terms of,

00:04:47.542 --> 00:04:48.982
are also SI units,

00:04:48.982 --> 00:04:50.440
and if we convert pascals

00:04:50.440 --> 00:04:52.605
to kilograms, meters, and seconds,

00:04:52.605 --> 00:04:53.935
we get that one pascal

00:04:53.935 --> 00:04:58.673
is one kilogram per meter seconds squared.

00:04:59.732 --> 00:05:01.585
What this tells us is that,

00:05:01.585 --> 00:05:05.116
we have to multiply
this by units of volume,

00:05:05.116 --> 00:05:07.163
and whatever we multiply it by

00:05:07.163 --> 00:05:10.089
should give us units of joules.

00:05:10.089 --> 00:05:11.726
So, what we need to do here,

00:05:11.726 --> 00:05:15.273
is convert our liters to meters cubed.

00:05:17.455 --> 00:05:18.742
And if we do that,

00:05:18.742 --> 00:05:22.840
everything is in terms of
kilograms, meters, and seconds,

00:05:22.840 --> 00:05:25.511
this meters cancels out with one of these,

00:05:25.511 --> 00:05:27.761
and we end up with one
kilogram meters squared

00:05:27.761 --> 00:05:29.916
per second squared on both sides.

00:05:29.916 --> 00:05:32.916
So then, everything is in terms of joules.

00:05:32.916 --> 00:05:34.566
That's not the only way
that you could've made sure

00:05:34.566 --> 00:05:36.458
that the units made sense.

00:05:36.458 --> 00:05:38.780
You could've converted
them to something else,

00:05:38.780 --> 00:05:40.080
but basically you just have to make sure

00:05:40.080 --> 00:05:43.457
that all of the units you're
using in your equation

00:05:43.457 --> 00:05:45.247
match each other if
you're going to add them

00:05:45.247 --> 00:05:46.257
or subtract them.

00:05:46.257 --> 00:05:48.507
All of this is to say
that we need to make sure

00:05:48.507 --> 00:05:50.999
we convert liters to meters cubed,

00:05:50.999 --> 00:05:54.406
so that everything works
out in terms of joules.

00:05:54.406 --> 00:05:55.184
So if we do that,

00:05:55.184 --> 00:05:57.977
we get that minus 485 joules,

00:05:59.376 --> 00:06:03.167
1.01 times ten to the 5th pascals,

00:06:04.241 --> 00:06:07.558
times negative .25 liters,

00:06:07.558 --> 00:06:09.995
which is the change in
volume, which is negative,

00:06:09.995 --> 00:06:10.800
the volume went down,

00:06:10.800 --> 00:06:12.797
so the change in volume
should be negative,

00:06:12.797 --> 00:06:16.303
and then we have to
add one more thing here

00:06:16.303 --> 00:06:19.816
to convert our liters to meters cubed.

00:06:21.643 --> 00:06:24.158
So, one liter is equal to one,

00:06:25.478 --> 00:06:29.156
ten to the minus third, meters cubed.

00:06:30.328 --> 00:06:32.127
So now our liters cancel out,

00:06:32.127 --> 00:06:35.960
and pascals times meters
cubed, gives us joules.

00:06:35.960 --> 00:06:38.874
So that gives us, that delta U,

00:06:38.874 --> 00:06:40.279
or change in internal energy,

00:06:40.279 --> 00:06:43.599
is negative 485 joules,

00:06:43.599 --> 00:06:45.596
then if we plug this
all into our calculator

00:06:45.596 --> 00:06:46.919
to calculate the work,

00:06:46.919 --> 00:06:51.919
we get positive 25.25 joules.

00:06:53.457 --> 00:06:55.999
So if we add our heat and our work here,

00:06:55.999 --> 00:06:57.821
we get that the overall change

00:06:57.821 --> 00:07:00.250
in internal energy for this process

00:07:00.250 --> 00:07:03.463
is negative 460 joules.

00:07:04.636 --> 00:07:06.378
So, the key things to remember here,

00:07:06.378 --> 00:07:07.736
for this kind of problem,

00:07:07.736 --> 00:07:10.802
is to double check your
signs for work and heat,

00:07:10.802 --> 00:07:13.238
and also to make sure
all of your units match.


Computing the partial derivative of a vector-valued function

WEBVTT
Kind: captions
Language: en

00:00:00.422 --> 00:00:01.360
- [Voiceover] Hello, everyone.

00:00:01.360 --> 00:00:02.886
So what I'd like to do
here and in the following

00:00:02.886 --> 00:00:05.136
few videos is talk about how you take the

00:00:05.136 --> 00:00:08.529
partial derivative of
vector valued functions.

00:00:08.529 --> 00:00:10.448
So the kind of thing I have in mind

00:00:10.448 --> 00:00:12.957
will be a function with
a multi-variable input,

00:00:12.957 --> 00:00:16.333
so this specific example have
a two variable input, p and s.

00:00:16.333 --> 00:00:18.101
You could think of that
as a two-dimensional space

00:00:18.101 --> 00:00:21.091
as the input or just two separate numbers.

00:00:21.091 --> 00:00:23.862
And its output will be three-dimensional.

00:00:23.862 --> 00:00:27.158
The first component, p
squared minus s-squared.

00:00:27.158 --> 00:00:30.575
The y component will be s times t.

00:00:31.586 --> 00:00:34.973
And that z component
will be t times s-squared

00:00:34.973 --> 00:00:39.140
minus s times t-squared,
minus s times t-squared.

00:00:40.295 --> 00:00:42.980
And the way that you
compute a partial derivative

00:00:42.980 --> 00:00:46.548
of a guy like this, is actually
relatively straight-forward.

00:00:46.548 --> 00:00:48.452
If you're to just guess
what it might mean,

00:00:48.452 --> 00:00:50.715
you'll probably guess right.

00:00:50.715 --> 00:00:53.443
It will look like partial
of v with respect to

00:00:53.443 --> 00:00:56.268
one of its input variables,
and I'll choose t

00:00:56.268 --> 00:00:58.258
with respect to t.

00:00:58.258 --> 00:01:00.881
And you just do it component-wise,

00:01:00.881 --> 00:01:02.572
which means you look at each component

00:01:02.572 --> 00:01:04.209
and you with a partial derivative to that

00:01:04.209 --> 00:01:06.048
'cause each component is just a normal

00:01:06.048 --> 00:01:07.646
scaler valued function.

00:01:07.646 --> 00:01:10.054
So you go up to the top one and you say

00:01:10.054 --> 00:01:11.566
t-squared looks like a variable,

00:01:11.566 --> 00:01:15.348
as far t is concerned,
and this derivative is 2t.

00:01:15.348 --> 00:01:16.948
But s-squared looks like a constant,

00:01:16.948 --> 00:01:18.954
so its derivative is zero.

00:01:18.954 --> 00:01:22.298
S times t, when s looks like a constant

00:01:22.298 --> 00:01:23.987
and when t looks like a variable,

00:01:23.987 --> 00:01:26.196
has a derivative of s.

00:01:26.196 --> 00:01:29.487
Then t times s-squared,
when t's the variable

00:01:29.487 --> 00:01:32.322
and s is the constant, just
looks like that constant,

00:01:32.322 --> 00:01:35.892
which is s-squared
minus s times t-squared.

00:01:35.892 --> 00:01:38.393
So now a derivative of t-squared is 2t

00:01:38.393 --> 00:01:39.725
and that constant s stays in.

00:01:39.725 --> 00:01:42.725
So that two times s times t.

00:01:43.571 --> 00:01:45.895
And that's how you compute it,

00:01:45.895 --> 00:01:47.923
probably relatively straightforward.

00:01:47.923 --> 00:01:50.553
The way you do it with
respect to s is very similar,

00:01:50.553 --> 00:01:52.641
but where this gets fun
and where this gets cool

00:01:52.641 --> 00:01:55.271
is how you interpret the
partial derivative, right,

00:01:55.271 --> 00:01:57.697
how you interpret this
value that we just found.

00:01:57.697 --> 00:01:59.349
And what that means
depends a lot on how you

00:01:59.349 --> 00:02:01.290
actually visualize the function.

00:02:01.290 --> 00:02:03.204
So what I'll go ahead
and do in the next video

00:02:03.204 --> 00:02:06.341
and in the next few ones, is talk about

00:02:06.341 --> 00:02:07.174
visualizing this function.

00:02:07.174 --> 00:02:08.502
It'll be as a parametric surface

00:02:08.502 --> 00:02:09.776
and three-dimensional space.

00:02:09.776 --> 00:02:12.433
That's why I got my
grapher program out here

00:02:12.433 --> 00:02:13.988
and I think you'll find
there's actually a very

00:02:13.988 --> 00:02:17.109
satisfying understanding
of what this value means.


Partial derivative of a parametric surface, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.564 --> 00:00:01.714
- [Voiceover] Hello hello again.

00:00:01.714 --> 00:00:02.643
So, in the last video,

00:00:02.643 --> 00:00:04.756
I started talking about how you interpret

00:00:04.756 --> 00:00:08.123
the partial derivative of a
parametric surface function,

00:00:08.123 --> 00:00:10.514
of a function that has
a two variable input,

00:00:10.514 --> 00:00:13.440
and a three variable vector-valued output.

00:00:13.440 --> 00:00:15.495
And we typically visualize
those as a surface

00:00:15.495 --> 00:00:16.958
in three dimensional space,

00:00:16.958 --> 00:00:19.965
and the whole process,
I was saying, you think

00:00:19.965 --> 00:00:22.965
about how a portion of the t-s plane

00:00:24.202 --> 00:00:25.811
moves to that corresponding output.

00:00:25.811 --> 00:00:27.762
And again, I'm kind of
cheating with this animation,

00:00:27.762 --> 00:00:29.488
where, really, this isn't
the t-s plane, right?

00:00:29.488 --> 00:00:30.855
This is on the x-y plane.

00:00:30.855 --> 00:00:33.875
t-s plane should just be some
separate space over here,

00:00:33.875 --> 00:00:35.373
and we're imagining
moving that separate space

00:00:35.373 --> 00:00:37.149
over into three dimensions.

00:00:37.149 --> 00:00:40.597
But that's harder to animate,
so I'm just not gonna do it,

00:00:40.597 --> 00:00:44.764
and I'm gonna instead keep
things inside the x-y plane here.

00:00:45.815 --> 00:00:46.648
We're thinking about the squares

00:00:46.648 --> 00:00:50.165
being t and s ranging each from 0 to 3.

00:00:50.165 --> 00:00:53.183
And, what I said, for partial
derivative with respect to t,

00:00:53.183 --> 00:00:54.657
is you imagine the line

00:00:54.657 --> 00:00:57.432
that represents movement
in the t direction.

00:00:57.432 --> 00:00:59.661
You see how that line gets mapped

00:00:59.661 --> 00:01:02.401
as all of the points move to
their corresponding output.

00:01:02.401 --> 00:01:05.617
And the partial derivative vector

00:01:05.617 --> 00:01:07.918
gives you a certain tangent vector

00:01:07.918 --> 00:01:10.298
to the curve representing that line,

00:01:10.298 --> 00:01:13.003
which corresponds to
movement in the t direction.

00:01:13.003 --> 00:01:15.337
And the longer that is,
the faster the movement,

00:01:15.337 --> 00:01:19.087
The more sensitive it is to
nudges in the t direction.

00:01:19.087 --> 00:01:20.747
In the s direction,
let's say we were to take

00:01:20.747 --> 00:01:22.872
the partial derivative with respect to s.

00:01:22.872 --> 00:01:25.789
So I'll kind of clear this up here.

00:01:26.715 --> 00:01:28.793
Also clear up this guy.

00:01:28.793 --> 00:01:30.012
And if you said instead,

00:01:30.012 --> 00:01:32.694
"what if we were doing it
with respect to s," right?

00:01:32.694 --> 00:01:34.273
Partial derivative of v,

00:01:34.273 --> 00:01:36.676
the vector-valued function,
with respect to s.

00:01:36.676 --> 00:01:38.393
Well, you do something very similar.

00:01:38.393 --> 00:01:40.972
You would say "OK, What is
the line that corresponds

00:01:40.972 --> 00:01:43.468
"to movement in the s direction?"

00:01:43.468 --> 00:01:44.301
And the way I've drawn it,

00:01:44.301 --> 00:01:45.825
it's always going to be perpendicular,

00:01:45.825 --> 00:01:47.503
because we're in the t-s plane,

00:01:47.503 --> 00:01:50.557
the t axis is perpendicular
to the s plane.

00:01:50.557 --> 00:01:53.982
And in this case, this line
represents t = 1, right?

00:01:53.982 --> 00:01:55.584
You're saying t constantly equals 1,

00:01:55.584 --> 00:01:57.500
but you're letting s vary.

00:01:57.500 --> 00:01:59.555
And if you see how that line maps,

00:01:59.555 --> 00:02:01.865
as you move everything
from the input space

00:02:01.865 --> 00:02:05.325
over to the corresponding
points in the output space,

00:02:05.325 --> 00:02:07.821
that line tells you what happens

00:02:07.821 --> 00:02:11.861
as you're varying the
s value of the input.

00:02:11.861 --> 00:02:13.638
It kind of starts curving this way,

00:02:13.638 --> 00:02:15.820
and then it curves very much up and

00:02:15.820 --> 00:02:18.282
kind of goes off into the distance there.

00:02:18.282 --> 00:02:19.942
And again, the grid
lines here really help,

00:02:19.942 --> 00:02:22.461
because every time that you
see the grid lines intersect,

00:02:22.461 --> 00:02:25.120
one of the lines represents
movement in the t direction,

00:02:25.120 --> 00:02:28.046
and the other represents
movement in the s direction.

00:02:28.046 --> 00:02:30.995
And for partial derivatives,
we think very similarly.

00:02:30.995 --> 00:02:34.661
You think of that partial
s as representing-

00:02:34.661 --> 00:02:35.923
(voiced zooming sounds)

00:02:35.923 --> 00:02:38.033
Zoom on back here.

00:02:38.033 --> 00:02:40.158
That partial s you
think of as representing

00:02:40.158 --> 00:02:42.654
a tiny movement in the s direction,

00:02:42.654 --> 00:02:45.278
just a little smidge and nudge,

00:02:45.278 --> 00:02:47.542
somehow nudging that guy along.

00:02:47.542 --> 00:02:49.423
And then the corresponding nudge

00:02:49.423 --> 00:02:51.280
you look for in the output space,

00:02:51.280 --> 00:02:53.614
you say okay, if we nudge
the input that much,

00:02:53.614 --> 00:02:56.614
and we go over to the output, and...

00:02:58.428 --> 00:03:00.001
and maybe that tiny nudge

00:03:00.001 --> 00:03:02.638
corresponded with one
that's three times bigger.

00:03:02.638 --> 00:03:04.974
I don't know, but it looked
like it stretched things out.

00:03:04.974 --> 00:03:06.266
So that tiny nudge might
turn into something

00:03:06.266 --> 00:03:07.624
that's still quite small,

00:03:07.624 --> 00:03:10.062
but maybe three times bigger.

00:03:10.062 --> 00:03:11.363
But it's a vector.

00:03:11.363 --> 00:03:13.046
What you do is, you think of that vector

00:03:13.046 --> 00:03:15.647
as being your partial v,

00:03:15.647 --> 00:03:18.004
and you scale it by whatever the size

00:03:18.004 --> 00:03:19.943
of that partial s was, right?

00:03:19.943 --> 00:03:22.613
So the result that you get

00:03:22.613 --> 00:03:26.398
is a tangent vector that's
not puny, not a tiny nudge,

00:03:26.398 --> 00:03:30.538
but is actually a sizable tangent vector.

00:03:30.538 --> 00:03:32.801
And it's going to correspond to the rate

00:03:32.801 --> 00:03:35.739
at which changes, not just tiny changes,

00:03:35.739 --> 00:03:37.881
but the rate at which changes in s

00:03:37.881 --> 00:03:40.146
cause movement in the output space.

00:03:40.146 --> 00:03:42.143
So, let's actually
compute it for this case,

00:03:42.143 --> 00:03:46.659
just get some good
practice computing things.

00:03:46.659 --> 00:03:49.016
And if we look up here, the t value

00:03:49.016 --> 00:03:50.653
which used to be considered a variable

00:03:50.653 --> 00:03:52.069
when we were doing it with respect to t.

00:03:52.069 --> 00:03:53.915
But now that t value
looks like a constant,

00:03:53.915 --> 00:03:55.517
so its derivative is zero.

00:03:55.517 --> 00:03:58.037
Then -s², with respect to s,

00:03:58.037 --> 00:04:00.037
has a derivative of -2s.

00:04:02.147 --> 00:04:05.514
st: s looks like a variable,
t looks like a constant,

00:04:05.514 --> 00:04:08.370
the derivative is just that constant, t.

00:04:08.370 --> 00:04:11.168
Down here, ts²: t looks like a constant,

00:04:11.168 --> 00:04:13.301
s looks like a variable, so...

00:04:13.301 --> 00:04:14.801
2 times t times s.

00:04:16.229 --> 00:04:18.622
And then over here, we're subtracting off,

00:04:18.622 --> 00:04:21.081
s is the variable, t²
looks like a constant,

00:04:21.081 --> 00:04:22.498
so that constant.

00:04:25.226 --> 00:04:27.873
And let's say we plug in the value (1,1).

00:04:27.873 --> 00:04:30.532
This red dot corresponds to (1,1),

00:04:30.532 --> 00:04:32.946
so what we would get here,

00:04:32.946 --> 00:04:35.071
s is equal to 1, so that's -2,

00:04:35.071 --> 00:04:36.952
t is equal to 1, so that's 1,

00:04:36.952 --> 00:04:39.564
then 2 times 1 times 1, I'll write it.

00:04:39.564 --> 00:04:40.397
2*1*1

00:04:40.397 --> 00:04:41.230
minus 1²

00:04:43.397 --> 00:04:47.078
is gonna correspond to 1, that's 2-1.

00:04:47.078 --> 00:04:49.818
So what we would expect
for the tangent vector,

00:04:49.818 --> 00:04:51.478
the partial derivative vector,

00:04:51.478 --> 00:04:53.243
is the x-component should be negative,

00:04:53.243 --> 00:04:55.994
and then the y and z-components
should each be positive.

00:04:55.994 --> 00:04:57.597
And if we go over, and we take a look

00:04:57.597 --> 00:05:01.300
at what the movement along
the curve actually is,

00:05:01.300 --> 00:05:02.257
that lines up, right?

00:05:02.257 --> 00:05:05.782
Because, as you kind of
zip along this curve,

00:05:05.782 --> 00:05:10.124
you're moving to the
left, so the x-component

00:05:10.124 --> 00:05:12.155
of the partial derivative
should be negative.

00:05:12.155 --> 00:05:15.337
But you're moving upwards
as far as y is concerned,

00:05:15.337 --> 00:05:17.241
and you can also kinda see
that the leftward movement

00:05:17.241 --> 00:05:19.291
is kind of twice as fast
as the upward motion.

00:05:19.291 --> 00:05:21.836
The slope favors the x direction.

00:05:21.836 --> 00:05:23.940
And then as far as the z
component is concerned,

00:05:23.940 --> 00:05:25.739
you are, in fact, moving up.

00:05:25.739 --> 00:05:26.700
And maybe you could say,

00:05:26.700 --> 00:05:28.032
"Well, how do you know
which way you're moving,

00:05:28.032 --> 00:05:30.263
"are you moving that way, or is everything

00:05:30.263 --> 00:05:31.878
"switched the other way around?"

00:05:31.878 --> 00:05:35.477
And the benefit of animation here is,

00:05:35.477 --> 00:05:38.833
we can say, "Ah, as s is ranging from zero

00:05:38.833 --> 00:05:41.898
"up to three, this is the
increasing direction."

00:05:41.898 --> 00:05:44.463
And you just keep your eye
on what that direction is

00:05:44.463 --> 00:05:47.052
as we move things about.

00:05:47.052 --> 00:05:49.014
And that increasing direction
does kind of correspond

00:05:49.014 --> 00:05:51.197
with moving along the curve this way.

00:05:51.197 --> 00:05:54.959
So, you get a tangent
vector in the other way.

00:05:54.959 --> 00:05:56.898
And one kinda nice thing
about this, then, is

00:05:56.898 --> 00:06:00.775
the two different partial
derivative vectors that we found

00:06:00.775 --> 00:06:02.134
Each one of them, you could say,

00:06:02.134 --> 00:06:04.246
is a tangent vector to the surface, right?

00:06:04.246 --> 00:06:05.554
So the one that was a partial derivative

00:06:05.554 --> 00:06:07.923
with respect to t, over here,

00:06:07.923 --> 00:06:09.281
kinda goes in one direction,

00:06:09.281 --> 00:06:11.092
and the other one kinda
gives you a different notion

00:06:11.092 --> 00:06:15.458
of what a tangent vector
on the surface could be.

00:06:15.458 --> 00:06:18.836
And you could have a notion
of directional derivative too,

00:06:18.836 --> 00:06:21.065
that kind of combines
these in various ways,

00:06:21.065 --> 00:06:22.520
and that will get you
all the different ways

00:06:22.520 --> 00:06:26.069
that you can have a vector
tangent to the surface.

00:06:26.069 --> 00:06:28.693
And later on, I'll talk about
things like tangent planes,

00:06:28.693 --> 00:06:31.874
if you want to express
what a tangent plane is.

00:06:31.874 --> 00:06:34.475
And you kind of think
of that as being defined

00:06:34.475 --> 00:06:36.437
in terms of two different vectors.

00:06:36.437 --> 00:06:38.492
But for now, that's really
all you need to know

00:06:38.492 --> 00:06:42.915
about partial derivatives
of parametric surfaces.

00:06:42.915 --> 00:06:44.599
And the next couple
videos, I'll talk about

00:06:44.599 --> 00:06:47.072
what partial derivatives
of vector-valued functions

00:06:47.072 --> 00:06:48.468
can mean in other contexts,

00:06:48.468 --> 00:06:50.601
because it's not always
a parametric surface,

00:06:50.601 --> 00:06:52.134
and maybe you're not always thinking

00:06:52.134 --> 00:06:54.119
about a curve that could be moved along,

00:06:54.119 --> 00:06:56.418
but you still want to think,
"How does this input nudge

00:06:56.418 --> 00:06:57.834
"correspond to an output nudge,

00:06:57.834 --> 00:07:00.110
"and what's the ratio between them."

00:07:00.110 --> 00:07:01.903
So with that, I'll see you next video.