2d curl formula Khan Academy English Korean

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- [Voiceover] So after introducing the idea

벡터장에서의 유체 회전에 대한

of fluid rotation in a vector field like this,

아이디어를 소개했으니

let's start tightening up our grasp on this intuition

이제 우리가 얻은 직관을 구체화해서

to get something that we can actually apply formulas to.

공식에 적용해봅시다

So a vector field like the one that I had there,

다음과 같은 이차원 벡터장은

that's two-dimensional, is given by a function

이차원의 입력값과

that has a two-dimensional input

이차원의 출력값의

and a two-dimensional output.

함수로 나타낼 수 있습니다

And it's common to write the components of that output

출력값의 성분들은 일반적으로

as the functions p and q.

p,q라는 함수로 나타냅니다

So, each one of those p and q,

이 각각의 p와 q는

takes in two different variables as it's input.

2개의 변수를 입력값으로

가지게 됩니다

And what I want to do here, is talk about this idea of curl

이제 회전의 아이디어를 소개하겠습니다

and you might write it down as just curl,

curl of v, the vector field.

회전을 구해보겠습니다

Which takes in the same inputs that the vector field does.

회전도 벡터장과 같은 입력값을 갖습니다

And because this is the two-dimensional example,

그리고 이것이 2차원에서의 예시이므로

뒤에서 배울

just to distinguish it from three-dimensional curl,

3차원 회전과 구별하기 위해서

which is something we'll get later on, two d curl of v.

이차원 회전이라고 표기하겠습니다

So you're kind of thinking of this as a differential thing,

회전 함수를 미분 비슷하게 생각합시다

in the same way that you have, you know, a derivative, dx is gonna take in some kind of a function.

dx가 어떤 함수를 받아들이면

And you give it a function and it gives you a new function,

그것이 새로운 도함수라는 새로운 함수를

the derivative.

반환하는 것처럼요

Here, you think of this 2d curl, as like an operator,

2차원 회전을 작용자로 생각하면

you give it a function, a vector field function,

이것은 벡터장 함수를 받아들여서

and it gives you another function,

다른 함수를 반환하게 되는데

which in this case will be scalar valued.

회전함수는 스칼라 값을 반환합니다

And the reason it's scalar valued,

스칼라 값이 반환되는 이유는

is because at every given point,

모든 주어진 점에서

you want it to give you a number.

회전함수값이 수이기 때문입니다

So if I look back at the vector field,

여기 주어진

that I have here,

벡터장을 보면

we want, that at a point like this,

반시계 방향의 회전이

where there's a lot of counter-clockwise rotation

많이 일어나는 이런 점에서

happening around it, for the curl function

우리는 회전함수가

to return a positive number.

양수를 반환하기를 원합니다

But at a point like this, where there's some,

하지만 다음과 같은

where there's clockwise rotation happening around it,

시계방향의 회전이 있는 점에서는

you want the curl to return a negative number.

회전이 음수값을 반환하기를 원합니다

So, let's start thinking about what that should mean.

이것의 의미를 생각해 봅시다

And a good way to understand this

이차원 회전에 대해서

two-dimensional curl function

잘 이해하고

and start to get a feel for it,

감각을 키우기 위해서는

is to imagine the quintessential 2d curl scenario.

본질적인 이차원 회전상황을 상상해야합니다

Well let's say you have a point

여기 공간상에

and this here's going to be our point, xy,

x,y좌표의 점이

sitting of somewhere in space.

있다고 해봅시다

And let's say there's no vector attached to it,

여기에 연결된 벡터가 없다고 할 때

as in the values, p and q, and x and y, are zero.

p와 q는 x,y에서 0입니다

And then let's say that to the right of it,

오른쪽에는

you have a vector pointing straight up.

위쪽 방향의 벡터가 있다고 가정해봅시다

Above it, in the vector field,

위에서 벡터장은

you have a vector pointing straight to the left,

왼쪽 방향의 벡터를 가지고

to it's left, you have one pointing straight down,

왼쪽에서는 아래방향의 벡터

and below it, you have one pointing straight to the right.

아래에서는 오른쪽 방향의 벡터를 갖는다고 합시다

So in terms of the functions, what that means,

함수의 관점에서 생각해보면

is this vector, to it's right,

오른쪽의 벡터는

whatever point it's evaluated at,

그 점이 무엇이건

that's gonna be q is greater than zero.

Q값은 양수일 것입니다

So this function q, that corresponds to the y component,

그러므로 Q함수는 y 구성성분에 대응합니다

the up and down component of each vector,

즉 벡터의 위 아래 성분에 대응하므로

when you evaluate it at this point,

x,y 오른쪽의 점에서 Q의 값을

to the right of our xy point,

q's gonna be greater than zero.

양수가 될 것입니다

Where as if you evaluate it to the left over here,

반대로 왼쪽의 벡터의 Q값은

q would be less than zero, less than zero,

위의 예시에서는

in our kind of, perfect curl will be positive example.

0 이하일 것입니다

And then these bottom guys,

그리고 다음 예시의

if you start thinking about what this means for,

의미를 생각해보면

you'd have a rightward vector below,

오른쪽 벡터가 아래에 있고

and a leftward vector above,

왼쪽 벡터가 위에 있습니다

the one below it, whatever point you're evaluating that at,

아래에 있는 벡터는 무슨 점을 계산하던

p, which gives us the kind of,

p는 벡터의 좌우방향 성분을

left right component of these vectors,

since it's the first component of the output,

이것은 출력값의 첫번째 성분이므로

would have to be positive.

양수여야 합니다

And then above it, above it here,

그리고 그 위에는

when you evaluate p at that point,

p를 계산해보면

would have to be negative.

음의 값을 갖을 것입니다

Where as p, if you did it on the left and right points,

좌우의 점에서의 p값은

would be equal to zero because there's no x component.

x성분이 없으므로 0입니다

And similarly q, if you did it on the top and bottom points,

같은 방법으로 Q는 위와 아래점에서는

since there's no up and down component of those vectors,

벡터의 상하성분이 없으므로

would also be zero.

0일것입니다

So this is just the, the very specific,

이것은 그냥

almost contrived scenario that I'm looking at.

억지로 만들어낸 시나리오입니다

And I want to say, hey if this should have positive curl,

이것이 양의 회전의 예시라면

maybe if we look at the information,

우리가 만약 다음과 같은 경우에서

the partial derivative information to be specific,

about p and q, in a scenario like this,

편도함수를 구하면

it'll give us a way to quantify the idea of curl.

회전함수를 정량화하는데 도움이 될 것입니다

And first let's look at p.

먼저 p를 봅시다

So p starts positive, and as y increases,

p는 입력되는 y값이

as the y value of our input increases,

증가함에 따라

it goes from being positive to zero, to negative.

양수에서 0, 음수가 됩니다

So we would expect, that the partial derivative of p,

그러므로 우리는 p의 y에 대한

with respect to y,

편도함수를 생각해야합니다

so as we change that y component, moving up in the plane,

즉 y성분을 증가시키킬때

and look at the x component of the vectors,

벡터의 x성분은

that should be negative.

음수여야합니다

That should be negative in circumstances

양의 회전을 갖으면

where we want positive curl.

음의 값을 갖게 된다는 거지요

So all of this we're looking at cases,

우리는 양의 회전에 대한

you know the quintessential case where curl is positive.

경우를 생각하는 것이므로

So evidently, this is a fact,

that corresponds to positive curl.

양의 회전에 대응합니다

Where as q, let's take a look at q.

이제 Q를 봅시다

It starts negative, when you're at the left.

Q는 음수로 시작해서

And then becomes zero, then it becomes positive.

0, 양수로 증가합니다

So here, as x increases, q increases.

x가 증가함에 따라 Q또한 증가하지요

So we're expecting that a partial derivative of q,

즉, Q의 x에 대한 편도함수는

with respect to x, should be positive.

양수입니다

Or at the very least, the situations where,

다른 의미로는

the partial derivative of q with respect to x is positive,

Q의 x에 대한 편도함수가 양수이면

corresponds to positive two-dimensional curl.

그것이 양의 이차원 회전을 의미하는 것입니다

And in fact, it turns out,

이 사실을 종합하면

these guys tell us all you need to know.

회전에 대한 공식을 할 수 있습니다

We can say as a formula,

이차원 회전에 대한 공식으로

that the 2d curl, 2d curl, of our vector field v,

x,y에 대한 함수인 벡터장v에 대해서

as a function of x and y,

이차원 회전은

is equal to the partial derivative of q with respect to x. Partial derivative of q, with respect to x,

Q의 x에 대한 편도함수에서

and then I'm gonna subtract off the partial of p,

P의 y에 대한 편도함수를

with respect to y.

뺀 것과 같습니다

Because I want, when this is negative,

P의 편도함수가 음수일때

for that to correspond with more positive 2d curl.

양의 이차원 회전에 대응하므로

So I'm gonna subtract off, partial of p,

P의 y에 대한 편도함수를

with respect to y.

뺴주어야 합니다

And this right here,

이것이 바로

is the formula for two-dimensional curl.

이차원 회전의 공식입니다

Which basically, you can think of it as a measure,

이것은 공간의 어느 점에서든

at any given point you're asking,

그 점 주위의 정보가

how much does the surrounding information to that point,

얼마나 이 예시같은 반시계방향의 회전과

look like this set-up,

일치하는지에 대한

like this perfect counter-clockwise rotation set-up?

값을 알려주게 됩니다

And the more it looks like this set-up,

이 예시와 더 일치할수록

the more this value will be positive.

더 큰 양의 값을 갖고

And if it was the opposite of this,

반대의 경우, 즉

if each of the vectors was turned around

모든 벡터가 시계방향의

and you have clockwise rotation,

회전을 갖는다면

each of these values will become the negative

주어진 값들은 모두

of what it had been before.

음의 값을 갖게 될 것입니다.

So 2d curl would end up being negative.

이차원 회전값은 음수가 되는 것이죠

And in the next video, I'll show some examples

다음 영상에서는 이 공식을 사용하는

of what it looks like to use this formula.

다른 예시들을 소개하겠습니다

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