
English: 
- [Voiceover] Hello
everyone, so in this video
I'm gonna introduce vector fields.
Now these are a concept
that come up all the time
in multi variable calculus,
and that's probably because
they come up all the time in physics.
It comes up with fluid
flow, with electrodynamics,
you see them all over the place.
And what a vector field is,
is its pretty much a way
of visualizing functions
that have the same number
of dimensions in their
input as in their output.
So here I'm gonna write a function
that's got a two dimensional input
X and Y, and then its output
is going to be a two dimensional vector
and each of the components
will somehow depend on X and Y.
I'll make the first one
Y cubed minus nine Y
and then the second component,
the Y component of the output
will be X cubed minus nine X.
I made them symmetric here,
looking kind of similar
they don't have to be, I'm
just a sucker for symmetry.
So if you imagine trying to
visualize a function like this
with a graph it would be really hard

Korean: 
안녕하세요, 이 영상에서는
벡터장을 소개할까 합니다
다변수 미적분학의 단골 주제이고
물리학에서 자주 나오는
표현이죠
유체의 흐름과 전자기학에서
매우 자주 이용합니다
벡터장은 입력과 출력의
개수가 같은 함수를
시각화하는 방법입니다
입력이 2차원으로
x와 y이고
출력이 2차원이고
각 성분이 x와 y의 식인
함수를 하나 예로 들죠
첫 번째는 y^3-9y
두 번째 출력의 y성분은
x^3-9x로 합시다
대칭을 너무 좋아하다 보니 이렇게 썼지만
꼭 이런 식일 필요는 없습니다
이런 함수를 시각화하려 할 때
입력도 2차원이고

English: 
because you have two
dimensions in the input
two dimensions in the output
so you'd have to somehow
visualize this thing in four dimensions.
So instead what we do, we
look only in the input space.
So that means we look
only in the X,Y plane.
So I'll draw these coordinate axes
and just mark it up,
this here's our X axis
this here's our Y axis
and for each individual input point
like lets say one,two
so lets say we go to one,two
I'm gonna consider the
vector that it outputs
and attach that vector to the point.
So lets walk through an
example of what I mean by that
so if we actually evaluate F at one,two
X is equal to one Y is equal to two
so we plug in two cubed
whoops, two cubed
minus nine times two
up here in the X component
and then one cubed minus nine times Y
nine times one, excuse me

Korean: 
출력도 2차원이다 보니
그대로 표현하려면
4차원이니까
매우 어려울 것입니다
그래서 일단 입력공간만을 보자고요
xy평면만 다루면서
좌표축을 그리고
x축과 y축을
표시하고
각각의 입력점마다
(1, 2)라고 하죠
(1, 2)에서의
출력 벡터를 생각해서
그 점에서 뻗는 것입니다
결국 어떻게 되냐면
(1, 2)에서 함숫값을 구하면
x=1, y=2니까
2^3
 
-9*2가
x성분이고
1^3-9*1이
y성분이

English: 
down in the Y component.
Two cubed is eight nine times two is 18
so eight minus 18 is negative 10
negative 10
and then one cubed is one,
nine times one is nine
so one minus nine is negative eight.
Now first imagine that this was
if we just drew this vector where we count
starting from the origin,
negative one, two,
three, four, five, six,
seven, eight, nine, 10,
so its going to have
this as its X component
and then negative eight,
one, two, three, four,
five, six, seven, we're gonna
actually go off the screen
its a very very large vector
so its gonna be something here
and it ends up having
to go off the screen.
But the nice thing about vectors
it doesn't matter where they start
so instead we can start it
here and we still want it
to have that negative ten X component
and the negative eight, negative one, two,
three, four, five, six, seven, eight,
negative eight

Korean: 
되겠죠
2^3=8, 9*2=18이니
8-18=-10이군요
-10
그리고 1^3=1, 9*1=1이니
1-9=-8입니다
먼저 이 벡터에서
원점부터 하나씩 세서
1, 2
3, 4, 5, 6, 7, 8, 9, 10
이만큼이 x성분이고
-8은 1, 2, 3, 4
5, 6, 7, 8 화면 밖이네요
화면 밖으로
벗어날 만큼
큰 벡터입니다
벡터는 시점이 어디든
상관없으니까
여기서 시작해서 그대로
x성분이 -10이고
y성분 -8을, -1, 2,
3, 4, 5, 6, 7, 8
-8을

Korean: 
y성분으로 긋고
벡터장의 개념은
(1, 2)에서만 이렇게
표시하는 것이 아니라
많은 점에다 벡터를 표현하는 것이니
크기를 그대로 표현하면
매우 지저분해지겠군요
모두 표시하다 보면
이 점에서 이렇게 긴 벡터
이 점에서는 이렇게
아주 지저분해지겠죠
이것들을 모두 지우고
보통 크기를 줄여서
벡터의 실제 크기는
정확하지 않지만
각 점에서 어떤 방향인지
알게 합니다
이 그림에서 다른 수정은
원래 함숫값와 일치하게
그린 건 아니라는 거죠
모든 벡터가 같은 길이니까요
이 벡터를 단위 길이로
이것도 단위 길이로 바꿔서
실제 벡터의 길이는
함숫값에 따라 차이가 많이 나는데도

English: 
as its Y component there
and a plan with the vector field
is to do this at not just one,two
but at a whole bunch of different points
and see what vectors attach to them
and if we drew them all
according to their size
this would be a real mess.
There'd be markings all over the place
and this one might have some
huge vector attached to it
and this one would have some
huge vector attached to it
and it would get really really messy.
But instead what we do, just
gonna clear up the board here
we scale them down, this is common
you'll scale them down and
so that you're kind of lying
about what the vectors themselves are
but you get a much better feel for
what each thing corresponds to.
And another thing about this drawing
that's not entirely faithful
to the original function that we have
is that all of these
vectors are the same length.
I made this one just kind of the same unit
this one the same unit, and over here
they all just have the same length
even though in reality
the length of the vectors'

Korean: 
길이를 모두 같게 조정했습니다
벡터장을 그리거나
소프트웨어에서 그려질 때 자주 이렇게 하지만
보완하는 방법은 있습니다
한 방법은 벡터에 색을 칠해서
여기 다른 그림을 보죠
색깔이 길이를 나타내게 해서
단위 길이로 정리해서
깔끔해 보이면서도
붉고 따뜻한 색 벡터는
길이가 긴 것으로
푸른빛은 짧은 것으로 구별할 수 있습니다
다른 방법은 원래 길이의 배수로
짧게 줄이는 것으로
푸른 벡터들이
거의 길이가 0이 된 반면
붉은 벡터들은 길이가 거의 그대로죠
실제 함숫값에
대응하는 벡터는
아주 길 수 있기 때문에
보기 적당한 정도로
자주 줄입니다
다음 영상에서는 벡터장이 항상 활용된는
유체 흐름을 보여 드리고

English: 
output by this function
can be wildly different.
This is kind of common practice
when vector fields are drawn
or when some kind of software
is drawing them for you
so there are ways of getting around this
one way is to just use
colors with your vectors
so I'll switch over to a
different vector field here
and here color is used to
kind of give a hint of length
so it still looks organized
because all of them
have the same length but the difference
is that red and warmer
colors are supposed to
indicate this is a very
long vector somehow
and then blue would indicate
that its very short.
Another thing you can
do is scale them to be
roughly proportional
to what they should be
so notice all the blue vectors
scaled way down to basically be zero
red vectors kind of stay the same size
even though in reality this
might be representing a function
where the true vector
here should be really long
or the true vector should
be kind of medium length
its still common for people
to just shrink them down
so its a reasonable thing to view.
So in the next video I'm
gonna talk about fluid flow
a context in which vector
fields come up all the time

Korean: 
이는 벡터장이 있을 때
이를 직관적으로 이해하는
좋은 방법입니다

English: 
and its also a pretty
good way to get a feel for
a random vector field that you look at
to understand what its all about.
