# 微分算子

## 记号

${\displaystyle \sum _{k=0}^{n}c_{k}D^{k}}$

${\displaystyle \Delta =\nabla ^{2}=\sum _{k=1}^{n}{\partial ^{2} \over \partial x_{k}^{2}}}$

${\displaystyle \Theta =z{\mathrm {d} \over \mathrm {d} z}}$

${\displaystyle \Theta (z^{k})=kz^{k},\quad k=0,1,2,\dots }$

n个变量中齐次算子由

${\displaystyle \Theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}}$

## 一个算子的伴随

${\displaystyle Tu=\sum _{k=0}^{n}a_{k}(x)D^{k}u}$

${\displaystyle \langle Tu,v\rangle =\langle u,T^{*}v\rangle }$

### 单变量中的形式伴随

${\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)\,{\overline {g(x)}}\,\mathrm {d} x}$

${\displaystyle T^{*}u=\sum _{k=0}^{n}(-1)^{k}D^{k}[a_{k}(x)u]}$

### 多变量

${\displaystyle \langle f,P^{*}g\rangle _{L^{2}(\Omega )}=\langle Pf,g\rangle _{L^{2}(\Omega )}}$

### 例子

${\displaystyle Lu=-(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p)D^{2}u+(-p')Du+(q)u.\;\!}$

{\displaystyle {\begin{aligned}L^{*}u&{}=(-1)^{2}D^{2}[(-p)u]+(-1)^{1}D[(-p')u]+(-1)^{0}(qu)\\&{}=-D^{2}(pu)+D(p'u)+qu\\&{}=-(pu)''+(p'u)'+qu\\&{}=-p''u-2p'u'-pu''+p''u+p'u'+qu\\&{}=-p'u'-pu''+qu\\&{}=-(pu')'+qu\\&{}=Lu\end{aligned}}}

## 微分算子的性质

${\displaystyle D(f+g)=(Df)+(Dg)}$
${\displaystyle D(af)=a(Df)}$

${\displaystyle (D_{1}\circ D_{2},f)=D_{1}(D_{2}(f))}$

${\displaystyle Dx-xD=1}$

## 坐标无关描述以及与交换代数的关系

${\displaystyle i_{P}:J^{k}(E)\rightarrow F}$

${\displaystyle P={\hat {i}}_{P}\circ j^{k}}$

${\displaystyle [f_{k}[f_{k-1}[\cdots [f_{0},P]\cdots ]]=0}$

${\displaystyle [f,P](s)=P(f\cdot s)-f\cdot P(s)}$

## 注释

1. ^ 计算机科学高阶函数的方式
2. ^ 当然有理由不单限制于线性算子。例如在只考虑线性的情况下，是一个熟知的非线性算子。
3. ^ 参见二阶导数的对称性