# 二阶导数的对称性

${\displaystyle f(x_{1},x_{2},\dots ,x_{n})}$

${\displaystyle f_{ij}=f_{ji}}$

## 黑塞矩阵是典型对称的

f的二阶偏导数称为f黑塞矩阵主对角线之外的元素是混合导数；即关于不同两个变量相继之导数。

## 对称性的正式表述

${\displaystyle {\frac {\partial }{\partial x}}\left({\frac {\partial f}{\partial y}}\right)={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)}$

${\displaystyle \partial _{xy}f=\partial _{yx}f}$

Di . Dj = Dj . Di.

## 克莱罗定理

${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$

${\displaystyle \mathbb {R} ^{n}}$中任何一点 ${\displaystyle (a_{1},\dots ,a_{n})}$连续二阶偏导数，则对${\displaystyle \forall i,j\in \mathbb {N} \backslash \{0\}:i,j\leq n,}$

${\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(a_{1},\dots ,a_{n})={\frac {\partial ^{2}f}{\partial x_{j}\,\partial x_{i}}}(a_{1},\dots ,a_{n}).}$

### 克莱罗常数

${\displaystyle \sin({\widehat {\mathrm {A} }})={\Big |}\cos(\phi _{q})\sin({\widehat {\alpha }}_{q}){\Big |}.\,\!}$

## 对称性的要求

${\displaystyle f(x,y)={\begin{cases}{\frac {xy(x^{2}-y^{2})}{x^{2}+y^{2}}}&{\mbox{ for }}(x,y)\neq (0,0)\\0&{\mbox{ for }}(x,y)=(0,0).\end{cases}}}$

${\displaystyle \partial _{x}\partial _{y}f|_{(0,0)}=\lim _{\epsilon \rightarrow 0}{\frac {\partial _{y}f|_{(\epsilon ,0)}-\partial _{y}f|_{(0,0)}}{\epsilon }}=1}$

${\displaystyle f(h,k)-f(h,0)-f(0,k)+f(0,0)}$

[Di, Dj] = 0

## 参考文献

1. ^ James, R.C.（1966）Advanced Calculus. Belmont, CA, Wadsworth.