# 微分運算元

## 記號

${d \over dx}$
$D$，這裡關於哪個變量微分是清楚的，以及
$D_x$，這裡指明了變量。

$d^n \over dx^n$
$D^n$
$D^n_x$

$\sum_{k=0}^n c_k D^k$

$\Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}$

$\Theta = z {d \over dz}$

$\Theta(z^k) = k z^k,\quad k=0,1,2,\dots$

n個變量中齊次運算元由

$\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}$

## 一個運算元的伴隨

$Tu = \sum_{k=0}^n a_k (x) D^k u$

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

### 單變量中的形式伴隨

$\langle f, g \rangle = \int_a^b f (x) \, \overline{g (x)} \,dx$

$T^*u = \sum_{k=0}^n (-1)^k D^k [a_k (x)u]$

### 多變量

$\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}$

### 例子

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

\begin{align} 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{align}

## 微分運算元的性質

$D(f+g) =(Df)+(Dg)$
$D (af) = a (Df)$

$(D_1 \circ D_2,f) = D_1(D_2(f))$

$Dx - xD = 1$

## 坐標無關描述以及與交換代數的關係

$i_P: J^k (E) \rightarrow F$

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

$[f_k[f_{k-1}[\cdots[f_0,P]\cdots]]=0$

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