【高等数学】多元微分学（二）

隐函数的偏导数

二元方程的隐函数

$F(x,y)=0$

推出隐函数形式 $y=y(x)$. 欲求 $\frac{dy}{dx}$ 需要对 $F=0$ 两边同时对 $x$ 求全导

$0=\frac{d}{dx}F(x,y(x))=\frac{\partial F}{\partial x}\frac{dx}{dx}+\frac{\partial F}{\partial y}\frac{dy}{dx}=\frac{\partial F}{\partial x}\frac{dx}{dx}+\frac{\partial F}{\partial y}\frac{dy}{dx}$

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

三元方程（组）的隐函数

三元方程

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

推出隐函数形式 $z=z(x,y)$. 欲求 $\frac{\partial z}{\partial x}$ 需要对 $F=0$ 两边同时对 $x$ 求偏导

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

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

两个三元方程

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

推出隐函数形式 $z=z(x),y=y(x)$ 欲求 $\frac{dz}{dx}$ 需要对 $F=0,G=0$ 两边同时对 $x$ 求全导

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

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

根据克拉姆法则,

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

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

四元方程（组）的隐函数

四元函数方程

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

推出隐函数形式 $w=w(x,y,z)$

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

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

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

两个四元函数方程组

$F(x,y,z,w)=0$, $G(x,y,z,w)=0$ 推出隐函数形式 $z=z(x,y)$, $w=w(x,y)$

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

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

根据克拉姆法则, $J=\frac{\partial (F,G)}{\partial (z,w)}=\left|\begin{array}{cc}\frac{\partial F}{\partial z}& \frac{\partial F}{\partial w}\\ & \\ \frac{\partial G}{\partial z}& \frac{\partial G}{\partial w}\end{array}\right|$

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

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

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

三个四元函数方程组

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

推出隐函数形式 $y=y(x),z=z(x),w=w(x)$

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

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

$0=\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y}\frac{dy}{dx}+\frac{\partial H}{\partial z}\frac{dz}{dx}+\frac{\partial H}{\partial w}\frac{dw}{dx}$

由克拉姆法则,

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

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

五元方程

两个五元方程

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

推出隐函数形式 $u=u(x,y,z),v=v(x,y,z)$, 对 $F=0,G=0$ 同时对 $x$ 求偏微分

$0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial F}{\partial v}\frac{\partial v}{\partial x}$ $0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial G}{\partial v}\frac{\partial v}{\partial x}$

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

类似的

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

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

三个五元方程

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

推出隐函数形式 $z=z(x,y),u=u(x,y),v=v(x,y)$

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

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

$0=\frac{\partial H}{\partial x}+\frac{\partial H}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial H}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial H}{\partial v}\frac{\partial v}{\partial x}$

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

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

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

(选看) 反函数的偏导数

一元一维函数 $y=f(x)$

反函数为 $x=f^{-1}(y)=\varphi (y)$
${\varphi }^{\prime }(y)=\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}=\frac{1}{f^{\prime }(x)}=\frac{1}{f^{\prime }(\varphi (y))}$

二元二维函数 $u(x,y),v(x,y)$

若在 $x_0,y_0$ 处满足 雅可比行列式 $\frac{\partial (u,v)}{\partial (x,y)}\ne 0$， 则存在反函数 $x=x(u,v),y=y(u,v)$,
反函数的雅可比矩阵是原函数雅可比矩阵的逆

令
$F(x,y,u,v)=x-x(u(x,y),v(x,y))=0$
$G(x,y,u,v)=y-y(u(x,y),v(x,y))=0$
对 $F=0$ $G=0$ 同时对 $x$ 求偏导
$1=\frac{\partial x}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial x}{\partial v}\frac{\partial v}{\partial x}$
$0=\frac{\partial y}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial y}{\partial v}\frac{\partial v}{\partial x}$
对 $F=0$ $G=0$ 同时对 $y$ 求偏导
$0=\frac{\partial x}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial x}{\partial v}\frac{\partial v}{\partial y}$
$1=\frac{\partial y}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial y}{\partial v}\frac{\partial v}{\partial y}$

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

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

用线性代数矩阵的乘法表示

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

三元三维函数 $u=(x,y,z),v(x,y,z),w(x,y,z)$

类似地

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

(选看) 参数方程的微分

二维单参数方程 (一元二维)

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

三维单参数方程 (一元三维)

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

三维双参数方程 (二元三维)

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

推出隐函数 $s=s(x,y)$, $t=t(x,y)$

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

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

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

结合二元二维反函数

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

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

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