# 二階導數的對稱性

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

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

## 黑塞矩陣是典型對稱的

f的二階偏導數稱為f黑塞矩陣主對角線之外的元素是混合導數；即關於不同兩個變量相繼之導數。

## 對稱性的正式表述

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

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

Di . Dj = Dj . Di.

## 克萊羅定理

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

${\displaystyle \mathbb {R} ^{n}}$中任何一點 ${\displaystyle (a_{1},\dots ,a_{n}),}$連續二階偏導數，則對${\displaystyle \forall i,j\in \mathbb {N} \backslash \{0\}:i,j\leq n,}$

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

### 克萊羅常數

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

## 對稱性的要求

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

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

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

[Di, Dj] = 0