Editing Multi-index notation (section)

==Definition and basic properties==

An ''n''-dimensional '''multi-index''' is an <math display="inline">n</math>-[[tuple]]

:<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>

of [[non-negative integer]]s (i.e. an element of the ''<math display="inline">n</math>''-[[dimension]]al [[set (mathematics)|set]] of [[natural number]]s, denoted <math>\mathbb{N}^n_0</math>).

For multi-indices <math>\alpha, \beta \in \mathbb{N}^n_0</math> and <math>x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n</math>, one defines:

;Componentwise sum and difference
:<math>\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n)</math>
;[[Partial order]]
:<math>\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\{1,\ldots,n\}</math>
;Sum of components (absolute value) 
:<math>| \alpha | = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>
;[[Factorial]]
:<math>\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n!</math>
;[[Binomial coefficient]]
:<math>\binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots\binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!}</math>
;[[Multinomial coefficient]]
:<math display="block">\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} </math> where <math>k:=|\alpha|\in\mathbb{N}_0</math>.
;[[Power (mathematics)|Power]]
:<math>x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \ldots x_n^{\alpha_n}</math>.
;Higher-order [[partial derivative]]
:<math display="block">\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n},</math> where <math>\partial_i^{\alpha_i}:=\partial^{\alpha_i} / \partial x_i^{\alpha_i}</math> (see also [[4-gradient]]). Sometimes the notation <math>D^{\alpha} = \partial^{\alpha}</math> is also used.<ref>{{cite book |first=M. |last=Reed |first2=B. |last2=Simon |title=Methods of Modern Mathematical Physics: Functional Analysis I |edition=Revised and enlarged |publisher=Academic Press |location=San Diego |year=1980 |isbn=0-12-585050-6| page=319 }}</ref>