标签： 高等数学基础

$\{\begin{array}{l}\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\cdot \frac{\partial v}{\partial x}\\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u}\cdot \frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\cdot \frac{\partial v}{\partial y}\end{array}$

$\{\begin{array}{l}\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial x},\\ \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial y}.\end{array}$

$\frac{\mathrm{d}z}{\mathrm{d}x}$ $=$ $\frac{\partial z}{\partial u}$ $\cdot$ $\frac{\mathrm{d}u}{\mathrm{d}x}$ $+$ $\frac{\partial z}{\partial v}$ $\cdot$ $\frac{\mathrm{d}v}{\mathrm{d}x}$

函数 $z$ $=$ $f(x,y)$ 在点 $(x,y)$ 处可微 $\Rightarrow$ 偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 必存在.

偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 存在 $\Rightarrow$ 函数 $z$ $=$ $f(x,y)$ 在点 $(x,y)$ 处不一定可微.

偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 在点 $(x,y)$ 的某邻域内存在且连续 $\Rightarrow$ 函数 $z$ $=$ $f(x,y)$ 在点 $(x,y)$ 处可微.

$f_{xy}^{\prime \prime }(x,y)$ $=$ $f_{yx}^{\prime \prime }(x,y)$

若 $\underset{\rho \to 0}{lim}$ $\frac{\mathrm{\Delta }z-f_x^{\prime }(x,y)\mathrm{\Delta }x-f_y^{\prime }(x,y)\mathrm{\Delta }y}{\rho }$ $=$ $0$,

或者：

$\underset{\rho \to 0}{lim}$ $\frac{\mathrm{\Delta }z–\frac{\partial z}{\partial x}\mathrm{\Delta }x–\frac{\partial z}{\partial y}\mathrm{\Delta }y}{\rho }$ $=$ $0$,

则函数 $z$ $=$ $f(x,y)$ 可微.

$\mathrm{d}z$ $=$ $\frac{\partial z}{\partial x}$ $\mathrm{d}x$ $+$ $\frac{\partial z}{\partial y}$ $\mathrm{d}y$ $\Rightarrow$

$\mathrm{d}z$ $=$ $\frac{\partial z}{\partial x}$ $\mathrm{\Delta }x$ $+$ $\frac{\partial z}{\partial y}$ $\mathrm{\Delta }y$

二元函数 $z$ 的全增量 $\mathrm{\Delta }z$ 为：
$\frac{\partial z}{\partial x}$ $\mathrm{\Delta }x$ $+$ $\frac{\partial z}{\partial y}$ $\mathrm{\Delta }y$ $+$ $o(\varphi )$

二元函数 $z$ 在点 $(x,y)$ 处的微分为：
$\frac{\partial z}{\partial x}$ $\mathrm{\Delta }x$ $+$ $\frac{\partial z}{\partial y}$ $\mathrm{\Delta }y$

其中，$o(\varphi )$ 表示 $\varphi$ 的高阶无穷小.

$\frac{\partial z}{\partial y}$ $=$ $\underset{\mathrm{\Delta }\to y}{lim}$ $\frac{f(x,y+\mathrm{\Delta }y)–f(x,y)}{\mathrm{\Delta }y}$

$\frac{\partial z}{\partial x}$ $=$ $\underset{\mathrm{\Delta }\to x}{lim}$ $\frac{f(x+\mathrm{\Delta }x,y)-f(x,y)}{\mathrm{\Delta }x}$

$\{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\{\begin{array}{l}T(x,z)=0\\ y=0\end{array}$

$\{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}$ $\Rightarrow$ 消去 $x$ $\Rightarrow$ $\{\begin{array}{l}R(y,z)=0\\ x=0\end{array}$

$\{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}$ $\Rightarrow$ 消去 $z$ $\Rightarrow$ $\{\begin{array}{l}H(x,y)=0\\ z=0\end{array}$

$\frac{x^2}{a^2}$ $+$ $\frac{y^2}{a^2}$ $-$ $\frac{z^2}{b^2}$ $=$ $-1$