multiindex notation
Multiindices form a powerful notational device for keeping track of multiple^{} derivatives^{} or multiple powers. In many respects these resemble natural numbers^{}. For example, one can define the factorial^{}, binomial coefficients^{}, and derivatives for multiindices. Using these one can state traditional results such as the multinomial theorem, Leibniz’ rule, Taylor’s formula^{}, etc. very concisely. In fact, the multidimensional results are more or less obtained simply by replacing usual indices in $\mathbb{N}$ with multiindices. See below for examples.
Definition A multiindex is an $n$tuple $\alpha =({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})$ of nonnegative integers ${\alpha}_{1},\mathrm{\dots},{\alpha}_{n}$. In other words, $\alpha \in {\mathbb{N}}^{n}$. Usually, $n$ is the dimension^{} of the underlying space. Therefore, when dealing with multiindices, $n$ is usually assumed clear from the context.
Operations on multiindices
For a multiindex $\alpha $, we define the length (or order) as
$$\alpha ={\alpha}_{1}+\mathrm{\cdots}+{\alpha}_{n},$$ 
and the factorial as
$$\alpha !=\prod _{k=1}^{n}{\alpha}_{k}!.$$ 
If $\alpha =({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})$ and $\beta =({\beta}_{1},\mathrm{\dots},{\beta}_{n})$ are two multiindices, their sum and difference^{} is defined componentwise as
$\alpha +\beta $  $=$  $({\alpha}_{1}+{\beta}_{1},\mathrm{\dots},{\alpha}_{n}+{\beta}_{n}),$  
$\alpha \beta $  $=$  $({\alpha}_{1}{\beta}_{1},\mathrm{\dots},{\alpha}_{n}{\beta}_{n}).$ 
Thus $\alpha \pm \beta =\alpha \pm \beta $. Also, if ${\beta}_{k}\le {\alpha}_{k}$ for all $k=1,\mathrm{\dots},n$, then we write $\beta \le \alpha $. For multiindices $\alpha ,\beta $, with $\beta \le \alpha $, we define
$$\left(\genfrac{}{}{0pt}{}{\alpha}{\beta}\right)=\frac{\alpha !}{(\alpha \beta )!\beta !}.$$ 
For a point $x=({x}_{1},\mathrm{\dots},{x}_{n})$ in ${\mathbb{R}}^{n}$ (with standard coordinates) we define
$${x}^{\alpha}=\prod _{k=1}^{n}{x}_{k}^{{\alpha}_{k}}.$$ 
Also, if $f:{\mathbb{R}}^{n}\to \mathbb{R}$ is a smooth function^{}, and $\alpha =({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})$ is a multiindex, we define
$${\partial}^{\alpha}f=\frac{{\partial}^{\alpha }}{{\partial}^{{\alpha}_{1}}{e}_{1}\mathrm{\cdots}{\partial}^{{\alpha}_{n}}{e}_{n}}f,$$ 
where ${e}_{1},\mathrm{\dots},{e}_{n}$ are the standard unit vectors of ${\mathbb{R}}^{n}$. Since $f$ is sufficiently smooth, the order in which the derivations^{} are performed is irrelevant. For multiindices $\alpha $ and $\beta $, we thus have
$${\partial}^{\alpha}{\partial}^{\beta}={\partial}^{\alpha +\beta}={\partial}^{\beta +\alpha}={\partial}^{\beta}{\partial}^{\alpha}.$$ 
Examples

1.
If $n$ is a positive integer, and ${x}_{1},\mathrm{\dots},{x}_{k}$ are complex numbers, the multinomial expansion states that
$${({x}_{1}+\mathrm{\cdots}+{x}_{k})}^{n}=n!\sum _{\alpha =n}\frac{{x}^{\alpha}}{\alpha !},$$ where $x=({x}_{1},\mathrm{\dots},{x}_{k})$ and $\alpha $ is a multiindex. (proof (http://planetmath.org/MultinomialTheoremProof))

2.
Leibniz’ rule: If $f,g:{\mathbb{R}}^{n}\to \mathbb{R}$ are smooth functions, and $\beta $ is a multiindex, then
$${\partial}^{\beta}(fg)=\sum _{\alpha \le \beta}\left(\genfrac{}{}{0pt}{}{\beta}{\alpha}\right){\partial}^{\alpha}(f){\partial}^{\beta \alpha}(g),$$ where $\alpha $ is a multiindex.
References
 1 M. Reed, B. Simon, Methods of Mathematical Physics, I  Functional Analysis^{}, Academic Press, 1980.
Title  multiindex notation 

Canonical name  MultiindexNotation 
Date of creation  20130322 13:41:32 
Last modified on  20130322 13:41:32 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  15 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 0500 
Defines  multiindex 
Defines  multiindices 