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做設計網(wǎng)站模塊的網(wǎng)站,軟文代寫價格,web前端項目開發(fā)流程,外匯網(wǎng)站建設公司隱函數(shù)的偏導數(shù) 二元方程的隱函數(shù) F ( x , y ) 0 F(x,y)0 F(x,y)0 推出隱函數(shù)形式 y y ( x ) yy(x) yy(x). 欲求 d y d x \frac{d y}{d x} dxdy? 需要對 F 0 F0 F0 兩邊同時對 x x x 求全導 0 d d x F ( x , y ( x ) ) ? F ? x d x d x ? F ? y d y d x ? F…

隱函數(shù)的偏導數(shù)

二元方程的隱函數(shù)

$F (x, y) = 0$

推出隱函數(shù)形式 $y = y (x)$ . 欲求 $\frac{d y}{d x}$ 需要對 $F = 0$ 兩邊同時對 $x$ 求全導

$\fracvxwlu0yf4{dx} F(x,y(x))= \frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx}=\frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx}$

求出 $\frac{dy}{dx}= -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$

三元方程（組）的隱函數(shù)

三元方程

$F (x, y, z) = 0$ ,

推出隱函數(shù)形式 $z = z (x, y)$ . 欲求 $\frac{\partial z}{\partial x}$ 需要對 $F = 0$ 兩邊同時對 $x$ 求偏導

$0=\frac{\partial F}{\partial x} +\frac{\partial F}{\partial z} \frac{\partial z}{\partial x}$

求出 $\frac{\partial z}{\partial x}= -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$

兩個三元方程

$F (x, y, z) = 0$ , $G (x, y, z) = 0$

推出隱函數(shù)形式 $z = z (x), y = y (x)$ 欲求 $\frac{d z}{dx}$ 需要對 $F = 0, G = 0$ 兩邊同時對 $x$ 求全導

$0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}\frac{dy}{dx}+\frac{\partial F}{\partial z}\frac{dz}{dx}$

$0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial y}\frac{dy}{dx}+\frac{\partial G}{\partial z}\frac{dz}{dx}$

根據(jù)克拉姆法則,

$J=\frac{\partial (F,G)}{\partial(y,z)}= \left|\begin{array}{c c} \frac{\partial F}{\partial y} &\frac{\partial F}{\partial z} \\\\ \frac{\partial G}{\partial y} &\frac{\partial G}{\partial z}\end{array}\right|$

$\frac{d y}{d x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (x,z)}$ $\frac{d z}{d x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (y,x)}$

四元方程（組）的隱函數(shù)

四元函數(shù)方程

$F (x, y, z, w) = 0$ ,

推出隱函數(shù)形式 $w = w (x, y, z)$

$\frac{\partial F}{\partial x}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial x}$ 求出

$\frac{\partial w}{\partial x}= - \frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial w}}$ 類似的

$\frac{\partial w}{\partial y}= - \frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial w}}$ $\frac{\partial w}{\partial z}= - \frac{\frac{\partial F}{\partial z}}{\frac{\partial F}{\partial w}}$

兩個四元函數(shù)方程組

$F (x, y, z, w) = 0$ , $G (x, y, z, w) = 0$ 推出隱函數(shù)形式 $z = z (x, y)$ , $w = w (x, y)$

$0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z} \frac{\partial z}{\partial x}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial x}$

$0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial z} \frac{\partial z}{\partial x}+\frac{\partial G}{\partial w} \frac{\partial w}{\partial x}$

根據(jù)克拉姆法則, $J=\frac{\partial (F,G)}{\partial(z,w)}= \left|\begin{matrix}\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w}\\\\\frac{\partial G}{\partial z} &\frac{\partial G}{\partial w}\end{matrix}\right|$

$\frac{\partial z}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (x,w)}$ $\frac{\partial w}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (z,x)}$

類似的由 $0=\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z} \frac{\partial z}{\partial y}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial y}$ $0=\frac{\partial G}{\partial y}+\frac{\partial G}{\partial z} \frac{\partial z}{\partial y}+\frac{\partial G}{\partial w} \frac{\partial w}{\partial y}$ 得到

$\frac{\partial z}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (y,w)}$ $\frac{\partial w}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (z,y)}$

三個四元函數(shù)方程組

$F (x, y, z, w) = 0$ , $G (x, y, z, w) = 0$ , $H (x, y, z, w) = 0$ .

推出隱函數(shù)形式 $y = y (x), z = z (x), w = w (x)$

$0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} \frac{d y}{d x}+\frac{\partial F}{\partial z} \frac{d z}{d x}+\frac{\partial F}{\partial w} \frac{d w}{d x}$

$0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial y} \frac{d y}{d x}+\frac{\partial G}{\partial z} \frac{d z}{d x}+\frac{\partial G}{\partial w} \frac{d w}{d x}$

$0=\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y} \frac{d y}{d x}+\frac{\partial H}{\partial z} \frac{d z}{d x}+\frac{\partial H}{\partial w} \frac{d w}{d x}$

由克拉姆法則,

$\frac{\partial(F, G, H)}{\partial (y,z,w)} = \left| \begin{array}{c c c} \frac{\partial F}{\partial y} &\frac{\partial F}{\partial z} & \frac{\partial F}{\partial w} \\\\ \frac{\partial G}{\partial y}&\frac{\partial G}{\partial z} &\frac{\partial G}{\partial w} \\\\ \frac{\partial H}{\partial y}&\frac{\partial H}{\partial z} &\frac{\partial H}{\partial w}\end{array}\right|$

$\frac{d y}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (x,z,w)}$ $\frac{d z}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (y,x,w)}$ $\frac{d w}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (y,z,x)}$

五元方程

兩個五元方程

$F (x, y, z, u, v) = 0$ , $G (x, y, z, u, v) = 0$

推出隱函數(shù)形式 $u = u (x, y, z), v = v (x, y, z)$ , 對 $F = 0, G = 0$ 同時對 $x$ 求偏微分

$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial F}{\partial v}\frac{\partial v}{\partial x}$ $\frac{\partial G}{\partial x} + \frac{\partial G}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial G}{\partial v}\frac{\partial v}{\partial x}$

由克拉姆法則 $J=\frac{\partial(F,G)}{\partial (u,v)}$ , $\frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(x,v)}$ $\frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,x)}$

類似的

$\frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(y,v)}$ $\frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,y)}$

$\frac{\partial u}{\partial z}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(z,v)}$ $\frac{\partial u}{\partial z}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,z)}$

三個五元方程

$F (x, y, z, u, v) = 0$ , $G (x, y, z, u, v) = 0$ , $H (x, y, z, u, v) = 0$

推出隱函數(shù)形式 $z = z (x, y), u = u (x, y), v = v (x, y)$

$\frac{\partial F}{\partial x} +\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial F}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial F}{\partial v}\frac{\partial v}{\partial x}$

$\frac{\partial G}{\partial x} +\frac{\partial G}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial G}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial G}{\partial v}\frac{\partial v}{\partial x}$

$\frac{\partial H}{\partial x} +\frac{\partial H}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial H}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial H}{\partial v}\frac{\partial v}{\partial x}$

由克拉姆法則 $J=\frac{\partial(F,G,H)}{\partial (z,u,v)}$ ,

$\frac{\partial z}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(x,u,v)}$ $\frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,x,v)}$ $\frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,u,x)}$

類似的 $\frac{\partial z}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(y,u,v)}$ $\frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,y,v)}$ $\frac{\partial v}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,u,y)}$

(選看) 反函數(shù)的偏導數(shù)

一元一維函數(shù) $y = f (x)$

反函數(shù)為 $x=f^{-1}(y)=\phi(y)$
$\phi'(y)=\frac{dx }{dy}=\frac{1}{\frac{dy}{dx}}=\frac{1}{f'(x)}=\frac{1}{f'(\phi(y))}$

二元二維函數(shù) $u (x, y), v (x, y)$

若在 $x_0,y_0$ 處滿足雅可比行列式 $\frac{\partial (u,v)}{\partial (x,y)}\neq 0$ ，則存在反函數(shù) $x = x (u, v), y = y (u, v)$ ,
反函數(shù)的雅可比矩陣是原函數(shù)雅可比矩陣的逆

令
$F (x, y, u, v) = x ? x (u (x, y), v (x, y)) = 0$
$G (x, y, u, v) = y ? y (u (x, y), v (x, y)) = 0$
對 $F = 0$ $G = 0$ 同時對 $x$ 求偏導
$\frac{\partial x}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial x}{\partial v}\frac{\partial v}{\partial x}$
$\frac{\partial y}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial y}{\partial v}\frac{\partial v}{\partial x}$
對 $F = 0$ $G = 0$ 同時對 $y$ 求偏導
$\frac{\partial x}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial x}{\partial v}\frac{\partial v}{\partial y}$
$\frac{\partial y}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial y}{\partial v}\frac{\partial v}{\partial y}$

$J=\frac{\partial (x,y)}{\partial (u,v)}$

因此
$\frac{\partial u}{\partial x}=\frac{1}{J}\frac{\partial y}{\partial v}$
$\frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial y}{\partial u}$
$\frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial x}{\partial v}$
$\frac{\partial v}{\partial y}=\frac{1}{J}\frac{\partial x}{\partial u}$

用線性代數(shù)矩陣的乘法表示

$\left[\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix}\right]\left[\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{matrix}\right] =\left[\begin{matrix} 1 & 0\\ 0& 1 \end{matrix}\right]$

三元三維函數(shù) $u = (x, y, z), v (x, y, z), w (x, y, z)$

類似地

$\left[\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\\ \end{matrix}\right]\left[\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} &\frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} &\frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} &\frac{\partial w}{\partial z}\\ \end{matrix}\right] =\left[\begin{matrix} 1 & 0& 0\\ 0& 1 &0\\ 0& 0 &1 \end{matrix}\right]$

(選看) 參數(shù)方程的微分

二維單參數(shù)方程 (一元二維)

$x = x (t)$ , $y = y (t)$ ,
$\frac{d y}{d x}= \frac{\frac{d y}{d t}}{\frac{dx}{dt}}$

三維單參數(shù)方程 (一元三維)

$x = x (t)$ , $y = y (t)$ , $z = z (t)$
$\frac{d y}{d x}= \frac{\frac{d y}{d t}}{\frac{dx}{dt}}$
$\frac{d z}{d x}= \frac{\frac{d z}{d t}}{\frac{dx}{dt}}$
$\frac{d z}{d y}= \frac{\frac{d z}{d t}}{\frac{dy}{dt}}$

三維雙參數(shù)方程 (二元三維)

$x = x (s, t)$ , $y = y (s, t)$ , $z = z (s, t)$ , 求解 $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$

推出隱函數(shù) $s = s (x, y)$ , $t = t (x, y)$

$z = z (s, t) = z (s (x, y), t (x, y))$ ,

$\frac{\partial z}{\partial x}= \frac{\partial z}{\partial s}\frac{\partial s}{\partial x}+ \frac{\partial z}{\partial t}\frac{\partial t}{\partial x}$

$\frac{\partial z}{\partial y}= \frac{\partial z}{\partial s}\frac{\partial s}{\partial y}+\frac{\partial z}{\partial t}\frac{\partial t}{\partial y}$

結合二元二維反函數(shù)

記 $J=\frac{\partial (x,y)}{\partial (s,t)}$

$\frac{\partial s}{\partial x}=\frac{1}{J}\frac{\partial y}{\partial t}$
$\frac{\partial t}{\partial x}=-\frac{1}{J}\frac{\partial y}{\partial s}$
$\frac{\partial s}{\partial y}=-\frac{1}{J}\frac{\partial x}{\partial t}$
$\frac{\partial t}{\partial y}=\frac{1}{J}\frac{\partial x}{\partial s}$

$\frac{\partial z}{\partial x} = \frac{1}{J} \frac{\partial (z,x)}{\partial (s,t)}$
$\frac{\partial z}{\partial y} = \frac{1}{J} \frac{\partial (x,z)}{\partial (s,t)}$