
Korean: 
이전 영상에서
벡터장을 배웠고, 여기서는
그 활용 하나를 언급하려 합니다
좌표평면을 표시하고
작은 물방울들을 그리면
이것이 이리저리
이동합니다
이걸 수학적으로는 어떻게 나타낼까요?
모든 점에서 입자는
다른 방향으로 움직입니다
이쯤에서는 왼쪽 아래로 움직이고
여기서는 위로 빠르게
이쯤에서는 아래로 느리게 움직이죠
공간의 모든 점에 벡터를 배치하면
이것이 뭔가 유동의 성질을
나타낼 수도 있겠군요
확실하진 않지만
시선을 여기 한 점에 고정시키면
입자들이 지나갈 때마다
거의 같은 속도로 보이니
사실 유체 흐름이
시간에 의존해서
속도가 계속 바뀌는
경우도 있지만

English: 
- [Voiceover] So in the last video,
I talked about Vector fields,
and here I want to talk
about a special circumstance
where they come up.
So imagine that we're sitting
in the coordinate plane,
and that I draw for you a whole bunch
of little droplets, droplets
of water, and then these are
going to start flowing in some way.
How would you describe
this flow mathematically?
So at every given point, the particles
are moving in some different way.
Over here, they're kind of
moving down and to the left.
Here, they're moving kind of quickly up.
Over here, they're
moving more slowly down.
So what you might want
to do is assign a vector
to every single point in space,
and a common attribute of
the way that fluids flow.
This isn't necessarily
obvious, but if you look
at a given point in space,
let's say like right here,
every time that a particle
passes through it,
it's with roughly the same velocity,
so you might think over time that velocity
would change, and sometimes it does.
A lot of times there's some fluid flow
where it depends on
time, but for many cases

English: 
you can just say, at this point in space,
whatever particle is going through it,
it'll have this velocity vector.
So over here, they might
be pretty high upwards,
whereas here, it's kind of
a smaller vector downwards,
even though, I'll play the
animation a little bit more here,
and if you imagine doing this
at all of the different points
in space, and assigning
a vector to describe
the motion of each fluid
particle at each different point,
what you end up getting is a vector field.
So this here is a little
bit of a cleaner drawing
than what I have, and as I
mentioned in the last video,
it's common for these vectors
not to be drawn to scale,
but to all have the
same length, just to get
a sense of direction, and here you can see
each particle is flowing
roughly along that vector,
so whatever one it's closest to,
it's moving in that direction.
And this is not just a really
good way of understanding
fluid flow, but it goes
the other way around.
It's a really good way of understanding
vector fields themselves,
so sometimes you might

Korean: 
보통 이 점에서
입자가 지나갈 때
이 속도벡터를 가집니다
여기서는 위로 크게
여기서는 아래로 작게
애니메이션을 다시 틀죠
공간의 모든 점에 벡터를
그린 모습을 상상하면
각 지점에서 유체 입자의 움직임은
벡터장으로 나타낼 수 있습니다
좀 더 깔끔하게 그려 보면
지난 시간에서 보았듯이
벡터 크기가 같게
방향만 나타내게 조정이 되죠
각 입자가
그 벡터 근처를 지나갈 때
가장 가까운 벡터 방향으로
이동하게 됩니다
유체 입자를 벡터장으로 이해할 뿐만 아니라
반대로도 생각할 수 있습니다
벡터장 자체를 이해하는 방법으로
어떤 벡터장이 주어질 때

Korean: 
꼭 유체 흐름을
나타내지 않더라도
어떤 성질이 있고 대강 어떤 형태인지
이런 입자들을 가지고
어떻게 움직일지 예상하는 게
매우 유용한 방식입니다
예를 들어 이 벡터장의
애니메이션에서는
밀도가 바뀌지 않습니다
어떤 시점에서도 입자가
증가하거나 감소하지 않고
실제로 이것은
수학적 의미와 연결됩니다
나중에 발산이라는 주제를
공부하면 이해하게 되실 겁니다
이 벡터장을 보고
어떤 특성이 있는지 알고 싶다면
모든 방향으로 퍼져 나가는
유체를 생각해서
중앙에서는 밀도가 줄어들고
이것도 수학적 의미가 있습니다
다른 질문도 할 수 있겠죠
맨 처음에 본 것과 같은
이 유동을 보면

English: 
just be given some new vector field,
and to get a feel for what it's all about,
how to interpret it, what
special properties it might have,
it's actually helpful, even
if it's not meant to represent
a fluid, to imagine that
it does, and think of all
the particles, and think of
how they would move along in.
For example, this particular
one, as you play the animation,
as you let the particles
move along the vectors,
there's no change in the density.
At no point do a punch
of particles go inward,
or a bunch of particles go outward,
it stays kind of constant,
and that turns out
to have a certain mathematical
significance down the road.
You'll see this later on as we study
a certain concept called divergence.
And over here, you see this vector field,
and you might want to
understand what it's all about,
and it's kind of helpful
to think of a fluid
that pushes outward from everywhere,
and is kind of decreasing in
density around the center,
and that also has a certain
mathematical significance,
and it might also lead you to
ask certain other questions.
Like if you look at the fluid flow
that we started with in this video,

English: 
you might ask a couple
questions about it like
it seems to rotating around some points,
in this case counter
clockwise, but it's rotating
clockwise around others still.
Does that have any kind of
mathematical significance?
Does the fact that there
seem to be the same
number of particles roughly in this area,
but they're slowly spilling out there.
What does that imply for the function
that represents this whole vector field,
and you'll see a lot of this later on,
especially when I talk
about divergence and curl,
but here I just wanted to
give a little warmup to that
as we're just visualizing
multivariable functions.

Korean: 
몇 개의 점에서는
회전하는 것처럼 보입니다
여기서는 반시계 방향으로
여기서는 시계 방향으로요
이것이 수학적 의미를 가질까요?
대강 같은 수의 입자가
맴도는 것 같지만
조금씩 빠져나가고 있죠
이것이 벡터장으로 표현된
함수에서는 어떤 뜻일까요?
나중에 발산과 회전을 다룰 때
모두 이해하게 되겠지만
다변수함수의 시각화 강의이니까
일단 남겨 두죠
