Partial derivatives of vector fields Khan Academy English (auto-generated) Korean

Partial derivatives of vector fieldsYouTube↗ IA↗(long shot)
Howto & Style 30.8K views 181 likes 4 dislikes 24 May 2016

Khan Academy 9.4M subscribers United States YouTube↗

Click to Play Video

so let's start thinking about partial

벡터장의 편미분에

derivatives of vector fields so a vector

대해 알아봅시다

field is a function I'll do I'll just do

벡터장은 함수입니다

a two-dimensional example here there's

이차원의 예시를 들겠습니다

going to be something that has a two-dimensional input and then the

이차원의 입력을 가질 것입니다

output has the same number of dimensions

그리고 같은 차수의 출력이 있을 것입니다

that's the important part and each of

중요한 부분입니다

these components in the output is going

출력의 각각의 성분은

to depend somehow on the input variables

입력 변수에 다소 의존합니다

so the example have in mind will be x

times y is that first component and then

첫번째 성분이 xy이고

y squared minus x squared as that second component and you can compute the

두번째 성분이 y^2 - x^2 입니다

partial derivative of a guy like this

이것의 편미분은

right you'll take the partial derivative

계산할 수 있습니다

with respect to one of the input

입력한 변수 중 하나에

variables I'll choose X it's always a

대해 편미분을 취합니다

nice one to start with partial

x 로 합시다 x는 시작하기 편합니다

derivative with respect to X and if we

x 에 대한 편미분입니다

were to actually compute it in this case

이것을 계산하기 위해,

it's another function of x and y what

이것은 x와 y의 함수입니다

you do is you take the partial

즉, 성분별로

derivative component-wise so you go to

편미분을 합니다

each component and the first one you say

첫번째 성분을 해봅시다

okay X looks like a variable Y looks

해보겠습니다

like a constant the derivative will just

x는 변수입니다 y는 상수입니다

be that constant and then the partial

도함수는 상수인 y가 됩니다

derivative of this second component that

그리고 두번째 성분의 편미분을 해봅시다

Y squared looks like a constant

y^2 는 상수입니다

derivative of negative x squared with

- x^2 의 x에 대한 미분은

respect to X negative 2x so analytically

if you know how to take a partial

따라서 분석적으로,

derivative you already know how to take

만약 편미분을 취하는 방법을 안다면,

a partial derivative of vector valued

벡터장에 의한 벡터함수의

functions and hence vector fields but

편미분을 취하는 방법을 이미 알 것입니다

the fun part that the important part

여기서 중요한 것은,

here how do you actually interpret this

'이것을 실제로 어떻게 해석하는가?' 입니다

and this has everything to do with

이것은 어떤 방법으로의

visualizing it in some way so the the

시각화를 통해 알 수 있습니다

vector field the reason we call it a

벡터장을 벡터장이라 부르는 이유는

vector field is you kind of take the whole XY plane and you're gonna fill it

xy 평면 전체를 취하여

with vectors and concretely what I mean

이것을 벡터들로 채울 것이기 때문입니다

by that you'll take a given input what's

구체적으로,

an input you want to look at like um

주어진 투입을 취합니다 알고 싶은 투입이 무엇인가요?

I'll say maybe 1 2 or yeah let's do that

(1, 2) 라고 합시다

let's do 1 2 now which would mean you

그걸로 해보겠습니다 (1, 2 )입니다

kind of go x equals 1 and then y equals

x 값은 1이고

2 this input point and we want to

y 값은 2입니다

associate that with the output vector in

이것이 투이할 점입니다 그리고 이것을

some way and so let's just compute what

출력 벡터와 어떤 방법으로 연결지어야 합니다 무엇과 같은지 계산해봅시다

it should equal so when we plug in x equals 1 and y equals 2x times y becomes

x 값이 1이고 y 값이 2이면, xy 는 2가 됩니다

2 y squared minus x squared becomes 2

squared minus 1 squared so 4 minus 1 is

2^2 - 1^2으로

3 so we have this vector 2 3 that we

4 - 1 이 되어 3입니다 따라서 (2, 3) 벡터를 얻고

want to associate with that input point

이것이 투입할 점과 연결지은 것입니다

and vector fields you just attach the

벡터 장은 단지 두 점을 연결하면 됩니다

two points you just I'm going to take

(2, 3) 과 (1, 2) 를

the vector 2 three and attach it to this guy so we

연결하겠습니다

should have an X component of two and

x 성분은 2 이고

then a Y component of three so it's

y 성분은 3 입니다

going to end up looking something like

다음과 같이 생겼을 것입니다

let's see so a lot of component of three

y 성분이 3이므로

something like this so that'll be the

다음과 같습니다

vector and we attach it to that point

이것은 벡터가 되고 (1,2) 와 연결합니다

and in principle you do this to all of

원래는 이것을 서로 다른 점들에 대해 해야합니다

the different points and if you did what you'd get would be something like this

만약 했으면, 다음을 얻을 것입니다

and remember when we represent these

이들을 나타낼 때

especially with computers it tends to

특히 컴퓨터로 나타낼 때,

lie where each represented vector is

이것은 누워 있는 경향이 있고,

much much shorter than it should be in

실제보다 나타낸 벡터들은 훨씬 더 짧습니다

reality but you just want to squish them

그러나 이들을 같은 페이지에 놓아서

all on to the same page so they don't

가득 차지 않았으면 할 것입니다

overrun each other and here color is

여기서 색은 상대적인 길이의

supposed to give a general vague sense

일반적이고 모호한 감각을 줍니다

of relative length so ones that are blue

따라서 파란색 벡터는

should be thought of as much shorter

노란색 벡터보다 짧다고 생각됩니다

than the ones that are yellow but that doesn't really give like a specific

그러나 이것은 그들이 얼마나 길어야 하는지에

thought for how long they should be but

구체적인 생각을 주지는 않습니다

for partial derivatives we actually care

편미분에 대해서는

a lot about the specifics and if you

구체적이어야 합니다

think if you think back to how we

서로 다른 많은 맥락에서

interpret partial derivatives in a lot

어떻게 편미분을 해석하는지로

of other contexts what we want to do is

돌아가 봅시다

imagine this partial X here as a slight

x 방향으로 조금 이동하는

nudge in the X Direction right so this

x의 편미분을 할 것입니다

this was our original input and you

(1 , 2)는 투입이고,

might imagine just nudging it a little

이것을 조금 이동한다고 가정하는데

bit and the size of that nudge as a

이동한 크기는

number would be your partial X so then

편미분 x 입니다

the question is what's the resulting

따라서 문제는

change to the output and because the

'결과적으로 출력에 어떤 변화가 있는가?' 입니다

output is a vector the change in the

출력이 벡터이기 때문에,

output is also going to be a vector it's

출력의 변화 또한 벡터입니다

what we want is to say there's gonna be

따라서 우리가 할 것은

some other vector attached to this point

이 점에 붙여진

right it's gonna look very similar maybe

또다른 벡터입니다

it looks like maybe it looks something

그것은 원래 것과 매우 비슷합니다 다음과 같습니다

like this so something similar but maybe

아마 이런 식일 것입니다

a little bit different and you you want

조금 다르지만, 비슷합니다

to take that that difference in vector

벡터 v의 편미분을 구해야 합니다

form and I'll describe what I mean by

앞서 말한 것을 나타내 보겠습니다

that in just a moment and then divide it by the size of that original nudg so to

그리고 x의 편미분으로 나누겠습니다

be to be much more specific about what I

더 구체적으로,

mean here if you're comparing two

만약 시점이 서로 다른

different vectors and they're rooted in

서로 다른 두 벡터를 비교하면,

two different spots I think a good way

이것을 시작하는 좋은 방법은

to start is to just move them to a new

그들의 시점이 같도록

space where they're rooted in the same

새로운 공간으로 이동시키는 것입니다

spot so in this case I'm just gonna kind

이 경우에서,

of draw a separate space over here and

분리된 공간을 새로 그리겠습니다

be thinking of this as a place for these

두 벡터들이 있을

vectors to live and I'm gonna I'm gonna

공간을 생각해봅시다

put them both on this plane but I'm

둘다 이 평면에 놓는데

gonna route them each in the origin so

원점에서 시작하도록 놓겠습니다

this first one that has components 2 3

첫번째 것은 (2,3)의 성분을 가지는데,

let's give it a name right let's call

이것에 이름을 붙이겠습니다

this guy v1 ok so that'll be V 1 and

이 벡터를 V라고 합시다

then the nudged output the the second

이것이 벡터 V1입니다 그리고 이동시킨 출력을

one I'll call v2 vitu and let's say v2 is also in the

V2 라고 합시다

space and I might exaggerate the

V2 또한 이 평면에 있는데,

difference just so that we can see it

차이를 과장시키면,

here let's say let's say it was

다음과 같습니다

different in some way but in reality if

V2 는 V1와 다릅니다

it's a small nudge it'll be different in

만약 조금 이동시켰다면,

only a very small way now but let's say

아주 조금 다를 것입니다

let's say these were our two vectors the

우리의 두 벡터입니다

difference the difference between these

V1 과 V2의 차는

guys is going to be a vector that connects the tips and I'm going to call

종점을 연결하는 벡터입니다

that guy like partial V and the way that

이를 ∂V 라고 합시다

you can be thinking about this is to say

이것을 생각하는 방법은

that V 1 V 1 that original guy Plus that

원래 벡터인 V1에

that tiny nudge the difference between

작은 이동인 ∂V 를 더한 것은

them is equal to V 2

you know the nudged output and in terms

같은데, V2 는 이동된 출력입니다

of tip to tail with vectors either

벡터들의 종점에서 시점까지에

you're saying that kind of the green

관해서는 본 적이 있을 것입니다

vector plus that blue vector is the same

녹색 벡터 더하기 푸른색 벡터는

as that that pink vector that connects

분홍색 벡터와 같은데,

the the tail of the original one to the

이 벡터는 원래 벡터의 시점과

tip of the new one so when we're

새로운 벡터의 종점을 연결합니다

thinking of a partial derivative you're

편미분에 관해서 생각할 때,

basically saying hey what happens if we

기본적으로

take this the nudge the size of the

" 출력이동의 크기를

nudge of the output and then we divide

입력이동의 크기로

it by that nudge of the input so let's

나누면 어떤 일이 생길까?" 라고 말할 것입니다

say you were thinking of that original

입력이동의 크기를

not just being I don't know like of a size 1/2 like 0.5 as the the change in

0.5 라고 합시다 x 방향에서의 변화입니다

the X Direction then that would mean

그러면 분자에서 ∂V의

when you go over here and you say what's

의미는 무엇일까요?

that DV that changed changing vector V divided by DX you'd be dividing it by

0.5 and in principle you'd be thinking

∂V 를 0.5로 나누면

of that would mean that you're kind of

2를 곱한 것과 같으므로

scaling this by 2 as if to say this

이 벡터를 두 배한 것과 같습니다

little DV is 1/2 of of some other vector

이 작은 ∂V는 또다른 벡터의

and that other vector is what the

partial derivative is so this other

또다른 벡터는 편미분입니다

vector here the full blue guy would be

여기서 또다른 벡터 즉, 파란색의 긴 벡터는

DV you know scaled down or scaled up

∂V를 ∂x에 의해

however you want to think about it by

확대 혹은 축소된

that partial X and that's what makes it

벡터입니다

such that you know in principle if this

만약 ∂x 가 0.01 처럼

partial X change was really small was

매우 작은 값이라면,

like 1/100 and the output and I'd also

출력이동 또한

was really small as like 1/100 or you

0.01과 같이 매우 작을 것입니다

know something on that order it wouldn't

또는 그렇게 구체적이지

be specifically that then the DV DX that

않을 수도 있습니다 ∂V/∂x 는 여전히

change would still be a normal sized

일반적인 크기의 벡터 일 것입니다

vector and the direction that it points

그것이 가리키는 방향은

is still kind of an indication of the

당신이 이동함에 따라

direction that this green vector should

녹색 벡터가 변하는

change as you're as you're scooting over

방향과 같을 것입니다

so just to be concrete and you know actually compute this guy let's say we

구체적으로 계산하기 위해,

were to take this partial derivative

partial of V with respect to X

(1 , 2)에 대해

and evaluate it at that point one two

계산해 봅시다

that we're just dealing with one two

우리는 (1 , 2)를 다루고 있습니다

what that would mean Y is equal to two

y값이 2이므로

so that first component is 2 and then X

첫번째 성분은 2 입니다

is equal to 1 so that next one should be

그리고 x값은 1이므로

negative 2 and we can I guess we can see just how wrong my drawing was to start

두번째 성분은 -2 입니다 그림을 잘못

here I was just kind of guessing what

그렸습니다

the pink vector would be but I guess it

분홍색 벡터를 추측했는데,

changes in the direction of 2 negative 2

분홍색 벡터는

so that should be something you're all erase the what turns out to be the wrong

(2, -2)의 방향으로 변해야 합니다 다시 그려보겠습니다

direction here yeah but to do get rid of

여기가 잘못된 방향을 가리키고 있습니다

this guy and I guess the change should

이것을 제거하겠습니다

be in the direction kind of 2 as the X

새로운 변화는

component and then negative 2 that's a

x 성분이 2이고 y 성분이 -2인 점의

negative 2 as the Y component so the

방향입니다

derivative vector should look something

따라서 도함수 벡터는 이렇게 생겼을 것입니다

like this which means all corresponding

대응하는 모든 작은 ∂v의 이동은

little DV nudges will be slight changes

아주 작은 변화이고,

will be slight changes on that so these

이것의 아주 작은 변화입니다

will be your your DV is something in

이것은 ∂V입니다

that direction and what that means in

일단 파란색 벡터의 방향입니다

our vector field then as you move in the

벡터장에서 이것이 의미하는 것은,

X direction and consider the various

x 방향으로 이동함에 따라

vectors attached to each point as you

각각의 점에 붙여진 다양한 벡터를 고려하면

kind of passing through the point 1 2

(1 , 2) 를 지나가면서

the way that the vectors are changing

그 벡터들이 변화하는 벡터는

should somebody somehow you know down

오른쪽 아래를 향합니다

into the right the tip should move down

그 벡터의 종점은 오른쪽 아래로 이동해야 합니다

into the right so if this starts you know highly up and to the right then it

녹색 벡터가 높이 올라가고 오른쪽으로,

should be getting kind of shorter but

따라서 y축 방향으로는 줄고

then longer to the right so the v2 if I

x축 방향으로는 길어집니다

were to have drawn it more accurately

따라서 V2를

here you know what that nudged output

조금 더 정확하게 그리면, 이동한 출력은

should look like it would really be something that's kind of like I don't

다음과 같아야 합니다

know like this where it's getting

다음과 같아야 합니다

shorter in the Y direction but then

y축 방향으로는 짧아지지만,

longer in the X direction as per that

x축 방향으로는 길어집니다

blue nudging arrow and in the next video

파란색 이동벡터가 오른쪽 아래를 향함에 따라

I'll kind of go go through more examples

다음 영상에서

of how you might think of this how do

이것을 어떻게 생각하는가에 관한 더 많은 예시를 보이겠습니다

you think of it in terms of what each

각각의 성분의 의미의 측면에서 어떻게 생각하는지 말입니다

component means which becomes very important for later topics like

발산과 회전처럼

divergence and curl and I'll see you

나중 주제에 매우 중요하게 될 것입니다

다음 영상에서 봅시다

Similar videos:

Partial derivatives, introduction
Khan Academy
836.1K views 6.5K likes 294 dislikes 11 May 201632% match

Partial derivative of a parametric surface, part 2
Khan Academy
12.4K views 59 likes 2 dislikes 24 May 201632% match

Directional derivative
Khan Academy
387.4K views 2.9K likes 167 dislikes 11 May 201632% match

Partial derivative of a parametric surface, part 1
Khan Academy
23K views 125 likes 5 dislikes 24 May 201631% match

Partial derivatives of vector fields, component by component
Khan Academy
14.9K views 76 likes 3 dislikes 24 May 201630% match

Directional derivative, formal definition
Khan Academy
123.2K views 414 likes 16 dislikes 11 May 201629% match

Directional derivatives and slope
Khan Academy
98.4K views 446 likes 14 dislikes 11 May 201629% match

Partial derivatives and graphs
Khan Academy
138.8K views 572 likes 4 dislikes 11 May 201629% match

Gradient and contour maps
Khan Academy
122.6K views 424 likes 9 dislikes 11 May 201629% match

Formal definition of partial derivatives
Khan Academy
148.9K views 552 likes 19 dislikes 11 May 201628% match

Why the gradient is the direction of steepest ascent
Khan Academy
154.7K views 954 likes 41 dislikes 11 May 201628% match

Gradient and graphs
Khan Academy
189.1K views 1K likes 17 dislikes 11 May 201627% match

The Jacobian matrix
Khan Academy
172.8K views 854 likes 33 dislikes 10 Jan 201727% match

Controlling a plane in space
Khan Academy
30.6K views 141 likes 3 dislikes 15 Jun 201627% match

Computing a tangent plane
Khan Academy
55.9K views 249 likes 8 dislikes 15 Jun 201627% match

Curvature formula, part 4
Khan Academy
33.5K views 151 likes 5 dislikes 20 May 201627% match

Introduction to 3d graphs | Multivariable calculus | Khan Academy
Khan Academy
198.6K views 931 likes 24 dislikes 05 May 201626% match

3d vector fields, introduction | Multivariable calculus | Khan Academy
Khan Academy
119.1K views 477 likes 8 dislikes 05 May 201626% match

Multivariable chain rule intuition
Khan Academy
63.4K views 377 likes 12 dislikes 20 May 201626% match

Gradient
Khan Academy
636.3K views 5.4K likes 155 dislikes 23 May 201626% match

3d curl intuition, part 2
Khan Academy
19.6K views 133 likes 0 dislikes 26 May 201625% match

Vector fields, introduction | Multivariable calculus | Khan Academy
Khan Academy
190K views 876 likes 30 dislikes 05 May 201625% match

3d curl formula, part 1
Khan Academy
21.2K views 95 likes 0 dislikes 26 May 201625% match

Local linearization
Khan Academy
33K views 140 likes 4 dislikes 15 Jun 201625% match

Curvature formula, part 2
Khan Academy
58.3K views 240 likes 7 dislikes 20 May 201625% match

Settings Privacy About Contact Donate