# 多元隐函数的偏导数

1，方程 $$F(x,y,z)=0$$ 给出 $$z$$ 是 $$x,y$$ 的函数，则

$\frac{\partial z}{\partial x}=-\frac{F_x}{F_z},\quad \frac{\partial z}{\partial y}=-\frac{F_y}{F_z}$

$F_x+F_z\frac{\partial z}{\partial x}=0,\quad F_y+F_z\frac{\partial z}{\partial y}=0$

2，由方程组

\begin{cases}F(x,y,u,v)=0\\ G(x,y,u,v)=0\end{cases}

\begin{cases}F_x+F_u\frac{\partial u}{\partial x}+F_v\frac{\partial v}{\partial x}=0\\ G_x+G_u\frac{\partial u}{\partial x}+G_v\frac{\partial v}{\partial x}=0\end{cases}

\begin{cases}F_y+F_u\frac{\partial u}{\partial y}+F_v\frac{\partial v}{\partial y}=0\\ G_y+G_u\frac{\partial u}{\partial y}+G_v\frac{\partial v}{\partial y}=0\end{cases}

$\frac{\partial u}{\partial x}=\frac{\begin{vmatrix}-F_x&F_v\\ -G_x&G_v\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}},\qquad \frac{\partial v}{\partial x}=\frac{\begin{vmatrix}F_u&-F_x\\ G_u&-G_x\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}$

$\frac{\partial u}{\partial y}=\frac{\begin{vmatrix}-F_y&F_v\\ -G_y&G_v\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}},\qquad \frac{\partial v}{\partial y}=\frac{\begin{vmatrix}F_u&-F_y\\ G_u&-G_y\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}$

$F_x=y-z, F_y=x+z, F_z=y-x$

$\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}=-\frac{y-z}{y-x},\quad \frac{\partial z}{\partial y}=-\frac{F_y}{F_z}=-\frac{x+z}{y-x}$

\begin{array}{l}F_x=y^2z^3+3x^2x^2z-1\\ F_y=2xyz^3+2x^2yz-1\\ F_z=3xy^2z^2+x^3y^-1\end{array}

\begin{array}{l}\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}=-\frac{y^2z^3+3x^2x^2z-1}{3xy^2z^2+x^3y^-1}\\ \frac{\partial z}{\partial y}=-\frac{F_y}{F_z}=-\frac{2xyz^3+2x^2yz-1}{3xy^2z^2+x^3y^-1}\end{array}

$G_x=v,G_y=u,G_u=y,G_v=x$

\begin{align*}\frac{\partial u}{\partial x}&=\frac{\begin{vmatrix}-F_x&F_v\\ -G_x&G_v\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}=\frac{\begin{vmatrix}-u&-y\\ -v&x\end{vmatrix}}{\begin{vmatrix}x&-y\\ y&x\end{vmatrix}}=\frac{-xu-yv}{x^2+y^2}\end{align*}

$\frac{\partial v}{\partial x}=\frac{\begin{vmatrix}F_u&-F_x\\ G_u&-G_x\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}=\frac{\begin{vmatrix}x&-u\\ y&-v\end{vmatrix}}{\begin{vmatrix}x&-y\\y&x\end{vmatrix}}=\frac{-xv+yu}{x^2+y^2}$

$\frac{\partial u}{\partial y}=\frac{\begin{vmatrix}-F_y&F_v\\ -G_y&G_v\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}=\frac{\begin{vmatrix}v&-y\\ -u&x\end{vmatrix}}{\begin{vmatrix}x&-y\\y&x\end{vmatrix}}=\frac{xv-yu}{x^2+y^2}$

$\frac{\partial v}{\partial y}=\frac{\begin{vmatrix}F_u&-F_y\\ G_u&-G_y\end{vmatrix}}{\begin{vmatrix}F_u&F_v\\ G_u&G_v\end{vmatrix}}=\frac{\begin{vmatrix}x&v\\ y&-u\end{vmatrix}}{\begin{vmatrix}x&-y\\y&x\end{vmatrix}}=\frac{-xu-yv}{x^2+y^2}$