Gradient and contour maps Khan Academy English Korean

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- [Voiceover] So here I want to talk about

이번에는 그라디언트와

the gradient and the context of a contour map.

등고선도에 대해서 알아보겠습니다

So let's say we have a multivariable function. A two-variable function f of x,y.

두 개의 변수 x, y로 이루어진 다변수함수 f를 생각해 봅시다

And this one is just gonna equal x times y.

f(x, y) = xy입니다.

So we can visualize this with a contour map

우리는 이것을 xy 평면 위에 놓인

just on the xy plane.

등고선도로써 시각화할 수 있습니다

So what I'm gonna do

무슨 말이냐면

is I'm gonna go over here. I'm gonna draw a y axis

여기에 y축을 그리고

and my x axis. All right, so

x축을 그릴게요

this right here represents x values.

이것이 x값을 나타내고

And this represents y values.

이것이 y값을 나타냅니다

And this is entirely the input space.

그리고 이것 전체가 값을 입력받을 공간입니다

And I have a video on contour maps

참고로 등고선도에 관련된 영상이 따로 있으니

if you are unfamiliar with them

등고선도가 뭔지 잘 모른다면

or are feeling uncomfortable.

확인해 보시길 바랍니다

And the contour map for x times y

x 곱하기 y의 등고선도는

looks something like this.

다음과 같습니다

And each one of these lines represents a constant value.

각각의 선들은 함수가 어떤 상수값을 가질 때의 그래프를 나타냅니다

So you might be thinking that you have,

그래서 만약 여러분이

you know, let's say you want a

the constant value for f of x times y is equal to two.

f(x, y) = 2의 그래프를 찾고 싶다면

Would be one of these lines.

이 선 중 하나가

That would be what one of these lines represents.

그 그래프가 됩니다

And a way you could think about that

즉, 여러분이

for this specific function is you saying hey, when is x times y equal to two?

언제 x 곱하기 y가 2가 되는지 물어본다고 하면,

And that's kind of like the graph

사실 그것은

y equals two over x.

y = 2/x의 그래프를 찾는 것과 같습니다

And that's where you would see something like this.

그리고 그 그래프의 개형은 이 선들과 비슷한 것이 됩니다

So all of these lines,

그래서 이 모든 선들은

they're representing constant values for the function.

상수값을 가질 때의 함수를 나타내고 있습니다

And now I want to take a look at the gradient field.

이제는 그라디언트에 대해서 알아보고 싶습니다

And the gradient, if you'll remember,

그라디언트란,

is just a vector full of the partial derivatives of f.

f의 편도함수으로 이루어진 벡터값이죠

And let's just actually write it out.

한번 써보겠습니다

The gradient of f, with our little del symbol,

x와 y로 이루어진 함수 f에

is a function of x and y.

기호 델(∇)을 쓰면

And it's a vector-valued function

첫번째 좌표값은

whose first coordinate is

f를 x에 대해 편미분한 값이고

the partial derivative of f with respect to x.

두번째 좌표값은

And the second component

f를 y에 대해 편미분한

is the partial derivative with respect to y.

벡터 함수가 나오게 됩니다

So when we actually do this for our function,

그럼 한번 계산해 볼까요

we take the partial derivative with respect to x.

f를 x에 대해서 편미분해 봅시다

It takes a look.

x는 변수로 취급하고

X looks like a variable.

y는 상수로 취급합니다

Y looks like a constant.

따라서 xy를 x에 대해 편미분한 값은

The derivative of this whole thing is just equal to that constant, y.

그냥 상수 y입니다

And then kind of the reverse for when you

그리고 y에 대해서 편미분할 때에는

take the partial derivative with respect to y.

그냥 반대로 해주면 되죠

Y looks like a variable.

y는 변수 취급하고

X looks like a constant.

x는 상수 취급해 줍니다

And the derivative is just that constant, x.

따라서 xy를 y에 대해 편미분한 값은 그냥 상수 x입니다

And this can be visualized as a vector field

그리고 이 벡터값 역시

in the xy plane as well. You know, at every given point, xy,

xy 평면 위의 벡터장으로써 시각화할 수 있죠

so you kind of go like

x equals two, y equals one, let's say.

x = 2이고 y = 1일 때,

So that would be x equals two, y equals one.

그러니까 이 점이 (2, 1)입니다

You would plug in the vector

이 벡터에다가 x = 2, y = 1을 대입했을 때

and see what should be output.

나오게 되는 벡터값을

And at this point, the point is two, one.

이 점을 시점으로 하여

The desired output kind of swaps those.

그려주면 됩니다

So we're looking somehow to draw the vector one, two.

즉 벡터 [1, 2]를 다음과 같이

So you would expect to see the vector that has

x축으로 1만큼 가고

an x component of one and a y component of two.

y축으로 2만큼 가서 그려주면

Something like that.

이렇게 됩니다

But it's probably gonna be scaled down

하지만 우리는 벡터장을 그리고 있기 때문에

because of the way we usually draw vector fields.

벡터의 길이는 줄여질 겁니다

And the entire field looks like this.

최종적으로 전체 벡터장은 다음과 같이 생겼습니다

So I'll go ahead and erase what I had going on.

조금 더 깔끔히 보기 위해

Since this is a little bit clearer.

지금까지 한 것을 지우겠습니다

And remember, we scaled down all the vectors.

여느 벡터장처럼, 우리는 편의상 화살표의 길이를 줄인 대신,

The color represents length.

화살표의 색깔이 벡터의 길이를 나타냅니다

So red here is super-long.

즉 여기 빨간색은 엄청 길고

Blue is gonna be kind of short.

파란색은 조금 짧습니다

And one thing worth noticing.

한가지 주목할 만한 점은,

If you take a look at all of the given points around,

여기에 주어진 모든 점들을 보면,

if the vector is crossing a contour line,

만약 벡터가 등고선와 겹치면,

it's perpendicular to that contour line.

그 벡터는 등고선에 대해 수직입니다

Wherever you go.

어디든지요

You know, you go down here,

여기 아래쪽으로 가도,

this vector's perpendicular to the contour line.

이 벡터는 이 등고선에 대해 수직입니다

Over here, perpendicular to the contour line.

여기도 마찬가지고요

And this happens everywhere.

모든 점에서 그렇습니다

And it's for a very good reason.

여기에는 매우 합당한 이유가 있으며

And it's also super-useful.

매우 유용하기도 합니다

So let's just think about what that reason should be.

왜 이렇게 되는지 생각해 봅시다

Let's zoom in on a point.

한 점을 확대해 보겠습니다

So I'm gonna clear up our function here.

일단 이 함수를 지우고

Clear up all of the information about it.

이것들도 전부 지우고

And just zoom in on one of those points.

한 점을 확대해 볼게요

So let's say like right here.

이 점으로 합시다

We'll take that guy

and kind of imagine

이렇게 확대해 본다고 상상하면

zooming in and saying what's going on in that region?

여기에 어떤 일이 일어나고 있는 걸까요

So you've got some kind of contour line.

여기에 등고선이

And it's swooping down like this.

이렇게 지나간다고 합시다

And that represents some kind of value.

이 등고선은 어떤 상수값을 나타내죠

Let's say that represents the value f equals two.

이 등고선이 f = 2를 의미한다고 할게요

And, you know, it might not be a perfect straight line.

물론 이것은 완벽한 직선은 아니지만

But the more you zoom in,

확대하면 확대할수록,

the more it looks like a straight line.

이 선은 더욱더 직선과 가까워집니다

And when you want to interpret the gradient vector.

그라디언트에 대해서 해석할 때,

If you remember, in the video about how to interpret

저번 동영상에서 저는

the gradient in the context of a graph,

그라디언트는 그래프에서

I said it points in the direction of steepest descent.

가장 빠르고 가파른 증가 경로를 가르킨다고 했었습니다

So if you imagine all the possible vectors

그래서 만약 이 점으로부터

kind of pointing away from this point,

뻗어나가는 벡터들이 이렇게 있다고 할 때,

the question is, which direction should you move

어떤 방향으로 움직여야지

to increase the value of f the fastest?

f의 값이 가장 빠르게 증가할까요?

And there's two ways of thinking about that.

2가지 방법으로 생각해 볼 수 있습니다

One is to look at all of these different directions

한가지는 모든 방향들을 고려해 봐서

and say which one increases x the most?

어떤 방향이 가장 x가 빠르게 증가하는지를 찾아내는 것이고

But another way of doing it

다른 한가지 방법은

would be to get rid of them all

모든 방향들을 싹 다 없애고

and just take a look at another contour line

살짝만 증가하는 다른 등고선을

that represents a slight increase.

바라보는 것이죠

All right, so let's say

무슨 말이냐 하면,

you're taking a look at a contour line,

여기에 등고선이 있고

another contour line.

또다른 등고선이 있다고 합시다

Something like this.

And maybe that represents something that's right next to it.

그리고 이 등고선은 원래 등고선의 바로 곁에 있습니다

f = 2.1이라고 합시다

That represents, you know, another value that's very close.

그러니까 원래 등고선과 매우 가까운 값을 가집니다

And if I were a better artist,

만약 제가 조금 더

and this was more representative,

그림을 잘 그렸더라면

it would look like a line

이 두 선은 마치

that's parallel to the original one.

평행한 것처럼 보였을 겁니다

Because if you change the output

왜냐하면 원래 등고선으로부터

by just a little bit,

아주 조금만 이동하는 것은

the set of in points that look like it

원래 선들의 모든 점들이

is pretty much the same but just shifted over a bit.

아주 조금 평행이동한 것과 마찬가지이기 때문이죠

So another way we can think about the gradient here

여기서 그라디언트 벡터는

is to say of all of the vectors that move

f = 2인 등고선으로부터

from this output of two

f = 2.1인 등고선까지

up to the value of 2.1.

이동한 모든 벡터 중에

You know, you're looking at

가장 빠르게

all of the possible different vectors that do that. You know, which one does it the fastest?

이동한 벡터입니다

And this time, instead of thinking of the fastest

그리고 가장 빠르게 이동한 벡터는

as constant-length vectors,

이 벡터들 중에서

what increases it the most,

가장 짧게 이동한 벡터입니다

we'll be thinking, constant increase in the output.

일정 거리를 가장 짧은 거리로 이동한 벡터가

Which one does it with the shortest distance?

그 거리를 가장 빠르게 이동한 벡터겠지요

And if you think of them as being roughly parallel lines,

그리고 만약 이 두 선들이 평행하다고 가정하면

it shouldn't be hard to convince yourself

가장 짧은 거리는

that the shortest distance isn't gonna be,

you know, any of those.

It's gonna be the one that connects them

수직하게 잇는

pretty much perpendicular to the original line.

선일 것입니다

Because if you think about these as lines,

왜냐하면 이 선들은

And the more you zoom in,

확대하면 확대할수록

the more they pretty much look like parallel lines,

평행선에 가까워 지므로

the path that connects one to the other

서로를 잇는 가장 짧은 경로는

is gonna be perpendicular to both of them.

이 두 선을 수직으로 잇는 경로일 것입니다

So because of this interpretation of the gradient

즉 그라디언트 벡터는

as the direction of steepest descent,

가장 가파르고 빠른 증가 경로를 의미하는데,

it's a natural consequence

that every time it's on a contour line,

두 등고선으로부터 가장 짧은 거리

wherever you're looking

즉 등고선으로부터

it's actually perpendicular to that line.

수직 방향의 벡터가 됩니다

Because you can think about it

as getting to the next contour line as fast as it can.

다음 등고선으로 가장 빠르게 이동하는 방향은

Increasing the function as fast as it can.

함수를 가장 빠르게 증가시키는 방향과 똑같기 때문에죠

And this is actually a very useful intepretation

그리고 이것은 사실

of the gradient in different contexts.

그라디언트를 다양한 맥락에서 이해하는 아주 좋은 해석입니다

So it's a good one to keep in the back of your mind.

그러니 머릿속에 기억해 두고 계세요

Gradient is always perpendicular to contour lines.

그라디언트는 항상 등고선에 수직합니다

See you next video.

다음 비디오에서 봐요 :)

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