# 多重指标

## 定義與運算

$\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)$

$\alpha, \beta$ 為多重指標，定義：

$\alpha \pm \beta:= (\alpha_{1} \pm \beta_{1},\,\alpha_{2} \pm \beta_{2}, \ldots, \,\alpha_{n} \pm \beta_{n})$
$\alpha \le \beta \quad \Leftrightarrow \quad \alpha_{i} \le \beta_{i} \quad \forall\,i$
$| \alpha | = \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}$

$\alpha ! = \alpha_{1}! \cdot \alpha_{2}! \cdots \alpha_{n}!$
${\alpha \choose \beta} = \frac{\alpha!}{(\alpha - \beta)! \, \beta!}={\alpha_{1} \choose \beta_{1}}{\alpha_{2} \choose \beta_{2}}\cdots{\alpha_{n} \choose \beta_{n}}$ （假設 $\alpha \geq \beta$
$x = (x_1, \ldots, x_n)$，定義 $\mathbf{x}^\alpha = x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \ldots x_{n}^{\alpha_{n}}$
$D^{\alpha} := D_{1}^{\alpha_{1}} D_{2}^{\alpha_{2}} \ldots D_{n}^{\alpha_{n}}$ 其中 $D_{i}^{j}:=\part^{j} / \part x_{i}^{j}$

$D^i x^k = \begin{cases} \frac{k!}{(k-i)!} x^{k-i} & i\le k\\ 0 & i \nleq k \end{cases}$

## 應用

### 多元微積分

$s(\mathbf{x}) = \sum_I a_I \mathbf{x}^I$

$s(x_1, \ldots, x_n) = \sum_{i_1, \ldots, i_n} a_{i_1 \ldots i_n} x_1^{i_1} \cdots x_n^{i_n}$

$\left( \sum_{i=1}^{n}{x_i}\right)^k = \sum_{|\alpha|=k}^{}{\frac{k!}{\alpha!} \, \mathbf{x}^{\alpha}}$

$D^{\alpha}(uv) = \sum_{\nu \le \alpha}^{}{{\alpha \choose \nu}D^{\nu}u\,D^{\alpha-\nu}v}$

$f(\mathbf{x}+\mathbf{h}) = \sum_{|\alpha| \ge 0} \frac{D^{\alpha}f(\mathbf{x})}{\alpha !}\mathbf{h}^{\alpha}$

$f(\mathbf{x}+\mathbf{h}) = \sum_{|\alpha| \leq n}{\frac{D^{\alpha}f(\mathbf{x})}{\alpha !}\mathbf{h}^{\alpha}}+R_n(\mathbf{x},\mathbf{h})$

$R_n(\mathbf{x},\mathbf{h})= (n+1) \sum_{|\alpha| =n+1}\frac{\mathbf{h}^\alpha}{\alpha !}\int_0^1(1-t)^nD^\alpha f(\mathbf{x}+t\mathbf{h})\,dt$

### 偏微分算子

$P(D) = \sum_{|\alpha| \le N}{}{a_{\alpha}(x)D^{\alpha}}$

$\int_{\Omega}{}{u(D^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(D^{\alpha}u)v\,dx}$

## 文獻

• Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9