# 多重指标

## 定義與運算

${\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}$

${\displaystyle \alpha ,\beta }$ 為多重指標，定義：

${\displaystyle \alpha \pm \beta :=(\alpha _{1}\pm \beta _{1},\,\alpha _{2}\pm \beta _{2},\ldots ,\,\alpha _{n}\pm \beta _{n})}$
${\displaystyle \alpha \leq \beta \quad \Leftrightarrow \quad \alpha _{i}\leq \beta _{i}\quad \forall \,i}$
${\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}$

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

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

## 應用

### 多元微積分

${\displaystyle s(\mathbf {x} )=\sum _{I}a_{I}\mathbf {x} ^{I}}$

${\displaystyle s(x_{1},\ldots ,x_{n})=\sum _{i_{1},\ldots ,i_{n}}a_{i_{1}\ldots i_{n}}x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}}$

${\displaystyle \left(\sum _{i=1}^{n}{x_{i}}\right)^{k}=\sum _{|\alpha |=k}^{}{{\frac {k!}{\alpha !}}\,\mathbf {x} ^{\alpha }}}$

${\displaystyle D^{\alpha }(uv)=\sum _{\nu \leq \alpha }^{}{{\alpha \choose \nu }D^{\nu }u\,D^{\alpha -\nu }v}}$

${\displaystyle f(\mathbf {x} +\mathbf {h} )=\sum _{|\alpha |\geq 0}{\frac {D^{\alpha }f(\mathbf {x} )}{\alpha !}}\mathbf {h} ^{\alpha }}$

${\displaystyle f(\mathbf {x} +\mathbf {h} )=\sum _{|\alpha |\leq n}{{\frac {D^{\alpha }f(\mathbf {x} )}{\alpha !}}\mathbf {h} ^{\alpha }}+R_{n}(\mathbf {x} ,\mathbf {h} )}$

${\displaystyle R_{n}(\mathbf {x} ,\mathbf {h} )=(n+1)\sum _{|\alpha |=n+1}{\frac {\mathbf {h} ^{\alpha }}{\alpha !}}\int _{0}^{1}(1-t)^{n}D^{\alpha }f(\mathbf {x} +t\mathbf {h} )\,dt}$

### 偏微分算子

${\displaystyle P(D)=\sum _{|\alpha |\leq N}{}{a_{\alpha }(x)D^{\alpha }}}$

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

## 文獻

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