미분,differential



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1. Definition

$y=f(x)$ is a differentiable function.
The differential $dy$ is defined by the equation
$dy=f'(x)dx$
where differential $dx$ is an independent variable.

Note: $dy$ is a dependent variable; it depends on $x\textrm{ \& }dx.$

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2. Definition (Thomas)

Let $y=f(x)$ be a differentiable function.
The differential $dx$ is an independent variable.
The differential $dy$ is
$dy=f'(x)dx.$

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3. Definition (Zill)

디퍼렌셜 $dy$ 는
$dy=y'dx$
로 정의된다. i.e.
$y'=\frac{dy}{dx}$
(Advanced Engineering Math Zill 6e. 26p)

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4. Definition (utk.edu)

Definition. Let $y=f(x)$ be a differentiable function.
The differential of $x,\, dx,$ is an independent variable.
The differential of $y$ is defined as
$dy=f'(x) dx$

([http]http://archives.math.utk.edu/visual.calculus/3/differentials.2/index.html)

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5. Definition (Varberg)

Definition

Let $y=f(x)$ be a differentiable function of the independent variable $x.$

$\Delta x$ is an arbitrary increment in the independent variable $x.$
$dx,$ called the differential of the independent variable $x,$ is equal to $\Delta x.$
$\Delta y$ is the actual change in the variable $y$ as $x$ changes from $x$ to $x+\Delta x;$ that is, $\Delta y=f(x+\Delta x)-f(x).$
$dy,$ called the differential of the dependent variable $y,$ is defined by $dy=f'(x)dx.$

Derivative Rule Differential Rule
$\frac{dk}{dx}=0$ $dk=0$
$\frac{d(ku)}{dx}=k\frac{du}{dx}$ $d(ku)=kdu$
$\frac{d(u+v)}{dx}=\frac{du}{dx}+\frac{dv}{dx}$ $d(u+v)=du+dv$
$\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$ $d(uv)=udv+vdu$
$\frac{d(u/v)}{dx}=\frac{u(du/dx)-u(dv/dx)}{v^2}$ $d\left(\frac{u}{v}\right)=\frac{vdu-udv}{v^2}$
$\frac{d(u^n)}{dx}=nu^{n-1}\frac{du}{dx}$ $d(u^n)=nu^{n-1}du$
왼쪽에 $dx$ 를 곱하면 오른쪽 식이 됨을 볼 수 있음

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6. 부분적분 공식에 나오는 디퍼렌셜

곱의 미분법을 Leibniz식으로 써 보면,
$\frac{d(uv)}{dx}=\frac{du}{dx}v+u\frac{dv}{dx}$
$dx$ 를 곱하면,
$d(uv)=(du)v+u(dv)$
적분하면,
$uv=\int vdu+\int udv$
$\int udv=uv-\int vdu$
이렇게 부분적분,integration_by_parts 공식이 나옴

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7. Examples (misc)

$y=x^2$ 이고 $(x,y)\to(x+dx,y+dy)$ 를 가정, 그럼 $dy/dx$ 는 어떻게 구하는가?
$y+dy=(x+dx)^2$
$y+dy=x^2+2xdx+dx^2$
여기서 $dx^2$ 은 매우 작으므로 무시하고 양변에서 $y=x^2$ 을 cancel한다. 그러면
$dy=2xdx$
$\frac{dy}{dx}=2x$
// http://kocw.net/home/search/kemView.do?kemId=1215229 이종광 20m
$x=vy$
$dx=vdy+ydv$
$u=vx$
$u'=v'x+v$
$x$ 로 미분하면
$\frac{du}{dx}=\frac{dv}{dx}x+v$
$du=xdv+vdx=vdx+xdv$

독립변수가 두 개인 경우
$u=f(x,y)$
$du=f_x(x,y)dx+f_y(x,y)dy$
i.e.
$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$
(편미분,partial_derivative)

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8. 미소~, ~ 미분소, differential length/area/volume/etc.

기초 calculus에서 중적분(multiple integral)이 나올 때 쓰는 기호는
$dA=dxdy$
$dV=dxdydz$
라 하고
$\iint_A dA=\iint dydx$
$\iiint_V dV=\iiint_V dzdydx$
등을 보통 씀. Fubini's theorem에 의해 dx dy dz의 순서를 크게 신경쓰지 않아도 됨.
좌표계,coordinate_system에 따라 달라짐.

미소~ (ex. 미소변위)
~ 미분소 (ex.
선 미분소(differential line), 면적 미분소(differential area), 부피 미분소(differential volume) [1]
체적 미분소(differential volume) $dv(=dxdydz)$ , 면적 미분소(differential surface) $d\bar{a}$ [2]
길이 미분소(length differential) [3])

선미분소 : 방향을 가진 매우 작은 선분의 극한 [4]
$d\bar{l}=dx\hat{x}+dy\hat{y}+dz\hat{z}$

극좌표계, 원통좌표계의 미소면적/미소부피는 극좌표계,polar_coordinate_system 참조

구면좌표계의 경우는 [https]구좌표계에서의 미소부피 참조

(Ulaby: CHK, CLEANUP 특히 r, R을 각각 rho, r로 잘 바꾸었는지 check)

(rectangular coord system)
differential surface area: $d\vec{s}$
differential length: $dl_x, dl_y, dl_z$
$d\vec{\ell}$
$=\hat{x}dl_x+\hat{y}dl_y+\hat{z}dl_z$
$=\hat{x}dx+\hat{y}dy+\hat{z}dz$
differential area in the
y-z plane: $d\vec{s_x}=\hat{x}dl_y dl_z=\hat{x}dydz$
x-z plane: $d\vec{s_y}=\hat{y}dxdz$
x-y plane: $d\vec{s_z}=\hat{z}dxdy$
(cyl. coord system)
differential volume elements...........?
differential lengths along $\rho, \phi, z$ :
$dl_{\rho}=d\rho$
$dl_{\phi}=rd\phi$
$dl_z=dz$
differential length:
$d\vec{\ell}$
$=\hat{\rho}dl_{\rho}+\hat{\phi}dl_{\phi}+\hat{z}dl_z$
$=\hat{\rho}d\rho+\hat{\phi}\rho d\phi + \hat{z}dz$
differential surface area:
$d\vec{s_{\rho}}=\hat{\rho}dl_{\phi}dl_z=\hat{\rho}\rho d\phi dz$ (φ-z cylindrical surface),
$d\vec{s_{\phi}}=\hat{\phi}dl_{\rho}dl_z=\hat{\phi}d\rho dz$ (ρ-z plane),
$d\vec{s_z}=\hat{z}dl_{\rho}dl_{\phi}=\hat{z}\rho d\rho d\phi$ (ρ-φ plane).
differential volume:
$dv=dl_{\rho}dl_{\phi}dl_z=\rho d\rho d\phi dz$
이상 완벽하게 r->rho 바꾸었는지 모르겠음
(sph. coordinate system)
differential length $d\vec{\ell}$
$=\hat{r} dl_r + \hat{\theta}dl_{\theta} + \hat{\phi}dl_{\phi}$
$=\hat{r}dr + \hat{\theta}rd\theta + \hat{\phi}r\sin\theta d\phi$
vector differential surface $d\vec{s}$
$d\vec{s_r}$
$=\hat{r}dl_{\theta}dl_{\phi}$
$=\hat{r}r^2\sin\theta d\theta d\phi$ (θ-φ spherical surface),
$d\vec{s_{\theta}}$
$=\hat{\theta} dl_{r} dl_{\phi}$
$=\hat{\theta} r \sin\theta dr d\phi$ (r-φ conical surface),
$d\vec{s_{\phi}}$
$=\hat{\phi} dl_r dl_{\theta}$
$=\hat{\phi} r dr d\theta$ (r-θ plane)
differential volume $dv$
$dv=dl_r dl_{\theta} dl_{\phi} = r^2 \sin\theta dr d\theta d\phi$

(이상 완벽하게 R->r 바꾸었는지 chk)

(이상 Ulaby 5e, 틀린지 chk)

이하

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8.1. Cartesian, differential length

$\vec{dl}=dx\hat{x}+dy\hat{y}+dz\hat{z}$
for 선적분,line_integral of vector fields

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8.2. Cartesian, differential volume

$dv=dxdydz$
for 체적적분,volume_integral of vector fields

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8.3. Cartesian, differential surface

$\vec{ds_1}=\hat{n}ds_1$ where $\hat{n}=\hat{z},\; ds_1=dxdy,\; \vec{ds_1}=\hat{z}dxdy$
$\vec{ds_2}=\hat{y}dxdz$
for 면적분,surface_integral of vector fields

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8.4. 미분전하 differential charge

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9. Subdifferential

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10. Links

삼각함수의 미분 d(sinθ)등에 대한 동영상 (차동우)
https://www.youtube.com/watch?v=aucbh027j70
$d(\sin\theta)=\cos\theta d\theta$
$d(\cos\theta)=-\sin\theta d\theta$
$\sin(d\theta)=d\theta$
$\cos(d\theta)=1$

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11. Calculus Made Easy №15(장?)


$y=\sin\theta$ 인데 $\frac{d(\sin\theta)}{d\theta}$ 를 investigate.
$y+dy=\sin(\theta+d\theta)$
$dy=\sin(\theta+d\theta)-\sin\theta$
우변은 $\sin M-\sin N=2\cos\frac{M+N}{2}\sin\frac{M-N}{2}$ 식에 따라 변형하고, $M=\theta+d\theta,\,N=\theta$ 이면
$dy=2\cos\frac{\theta+d\theta+\theta}{2}\sin\frac{\theta+d\theta-\theta}{2}$
$dy=2\cos\left(\theta+\frac12d\theta\right)\sin\left(\frac12 d\theta\right)$
여기서 $d\theta$ 를 indefinitely small하다고 regard하면, $\theta$ 와 비교해 $\frac12d\theta$ 를 무시할 수 있고, $\sin\left(\frac12 d\theta\right)$ 를 $\frac12d\theta$ 와 같다고 볼 수 있다. 그러면
$dy=2\cos\theta\cdot\frac12d\theta$
$dy=\cos\theta\cdot d\theta$
$\frac{dy}{d\theta}=\cos\theta$

다음에는 코사인,cosine의 미분.
$y=\cos\theta$
$\cos\theta=\sin\left(\frac{\pi}{2}-\theta)$
따라서
$dy=d\left(\sin\left(\frac{\pi}{2}-\theta\right)\right)$
$=\cos\left(\frac{\pi}{2}-\theta\right)\times d(-\theta)$
$=\cos\left(\frac{\pi}{2}-\theta\right)\times(-d\theta)$
$\frac{dy}{d\theta}=-\cos\left(\frac{\pi}{2}-\theta\right)$
$\frac{dy}{d\theta}=-\sin\theta$

이어서 탄젠트,tangent, 이계미분 나오는데 생략.

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12. 궁금 QQQ

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13. 기타


Sub:
완전미분,exact_differential
{
https://en.wikipedia.org/wiki/Exact_differential
완전미분이 없으면 불완전미분(inexact differential)
RR:완전미분exact_differential
}
완전미분방정식exact de
RR:완전미방exact_DE

미분형식,differential_form


선형근사,linear_approximation와 밀접한데 관계 적을 것 TBW

AKA 미분소(素), 디퍼렌셜

Compare:
미분,derivative (도함수)
미분,differentiation (도함수를 찾는 것)
무한소,infinitesimal - 비슷?
차분(difference) - see 차이,difference

Up:
미적분,calculus

Sub:
미분방정식,differential_equation
미분연산자,differentiation_operator (curr at RR 미분연산자,differentiation_operator )
전미분,total_differential
Kaehler_differential =,Kaehler_differential . Kaehler_differential <- German umlaut transliteration을 찾아보면 (ä → ae)라서 pagename 이렇게 해야 할 듯 한데, chk
{
Kähler differential
https://ko.wikipedia.org/wiki/켈러_미분
https://en.wikipedia.org/wiki/Kähler_differential
https://ja.wikipedia.org/wiki/ケーラー微分
https://ncatlab.org/nlab/show/Kähler differential
...
"Kähler differential"
Naver:Kähler differential
Ggl:Kähler differential
}

See also 디퍼렌셜예제가있는페이지

Related: Compare: 차이,difference(=차분)

Twins:
RR: 미분,differential
https://ncatlab.org/nlab/show/differential (hard)
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