# 多元函数基本定义

## 二重极限

${\displaystyle 0<{\sqrt {(x-x_{0})^{2}+(y-y_{0})^{2}}}<\delta }$

${\displaystyle \left|f(x,y)-C\right|<\varepsilon }$

${\displaystyle \lim _{x\to x_{0} \atop y\to y_{0}}f(x,y)=C}$

## 多元函数的连续性

${\displaystyle \lim _{x\to x_{0} \atop y\to y_{0}}f(x,y)=f(x_{0},y_{0})}$

# 求导法则

## 复合求导法

1.若函数f(x,y)可微，且x=${\displaystyle \phi }$(t)，y=${\displaystyle \varphi }$(t)都对t可导，则复合函数f(${\displaystyle \phi }$(t),${\displaystyle \varphi }$(t))也对t可导，且满足：

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}={\frac {\partial f}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial f}{\partial y}}{\frac {\partial y}{\partial t}}}$

2.若函数f(u,v)可微，且u=${\displaystyle \phi }$(x,y)，v=${\displaystyle \varphi }$(x,y)都对t可导，则复合函数f(${\displaystyle \phi }$(x,y),${\displaystyle \varphi }$(x,y))也对(x,y)存在偏导数，且满足：

${\displaystyle \left({\begin{matrix}{\frac {\partial f}{\partial x}}\\\\{\frac {\partial f}{\partial y}}\end{matrix}}\right)=\left({\begin{matrix}{\frac {\partial u}{\partial x}}&{\frac {\partial v}{\partial x}}\\\\{\frac {\partial u}{\partial y}}&{\frac {\partial v}{\partial y}}\end{matrix}}\right)\left({\begin{matrix}{\frac {\partial f}{\partial u}}\\\\{\frac {\partial f}{\partial v}}\end{matrix}}\right)}$