# multi-index notation

Multi-indices 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 multi-indices. Using these one can state traditional results such as the multinomial theorem, Leibniz’ rule, Taylor’s formula   , etc. very concisely. In fact, the multi-dimensional results are more or less obtained simply by replacing usual indices in $\mathbb{N}$ with multi-indices. See below for examples.

Definition A multi-index is an $n$-tuple $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ of non-negative integers $\alpha_{1},\ldots,\alpha_{n}$. In other words, $\alpha\in\mathbb{N}^{n}$. Usually, $n$ is the dimension   of the underlying space. Therefore, when dealing with multi-indices, $n$ is usually assumed clear from the context.

## Operations on multi-indices

For a multi-index $\alpha$, we define the length (or order) as

 $|\alpha|=\alpha_{1}+\cdots+\alpha_{n},$

and the factorial as

 $\alpha!=\prod_{k=1}^{n}\alpha_{k}!.$

If $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and $\beta=(\beta_{1},\ldots,\beta_{n})$ are two multi-indices, their sum and difference  is defined component-wise as

 $\displaystyle\alpha+\beta$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\beta_{1},\ldots,\alpha_{n}+\beta_{n}),$ $\displaystyle\alpha-\beta$ $\displaystyle=$ $\displaystyle(\alpha_{1}-\beta_{1},\ldots,\alpha_{n}-\beta_{n}).$

Thus $|\alpha\pm\beta|=|\alpha|\pm|\beta|$. Also, if $\beta_{k}\leq\alpha_{k}$ for all $k=1,\ldots,n$, then we write $\beta\leq\alpha$. For multi-indices $\alpha,\beta$, with $\beta\leq\alpha$, we define

 ${\alpha\choose\beta}=\frac{\alpha!}{(\alpha-\beta)!\beta!}.$

For a point $x=(x_{1},\ldots,x_{n})$ in $\mathbb{R}^{n}$ (with standard coordinates) we define

 $x^{\alpha}=\prod_{k=1}^{n}x_{k}^{\alpha_{k}}.$

Also, if $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is a smooth function  , and $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is a multi-index, we define

 $\partial^{\alpha}f=\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}e_{1}\cdots% \partial^{\alpha_{n}}e_{n}}f,$

where $e_{1},\ldots,e_{n}$ 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 multi-indices $\alpha$ and $\beta$, we thus have

 $\partial^{\alpha}\partial^{\beta}=\partial^{\alpha+\beta}=\partial^{\beta+% \alpha}=\partial^{\beta}\partial^{\alpha}.$

## Examples

1. 1.

If $n$ is a positive integer, and $x_{1},\ldots,x_{k}$ are complex numbers, the multinomial expansion states that

 $(x_{1}+\cdots+x_{k})^{n}=n!\sum_{|\alpha|=n}\frac{x^{\alpha}}{\alpha!},$

where $x=(x_{1},\ldots,x_{k})$ and $\alpha$ is a multi-index. (proof (http://planetmath.org/MultinomialTheoremProof))

2. 2.

Leibniz’ rule: If $f,g\colon\mathbb{R}^{n}\to\mathbb{R}$ are smooth functions, and $\beta$ is a multi-index, then

 $\partial^{\beta}(fg)=\sum_{\alpha\leq\beta}{\beta\choose\alpha}\partial^{% \alpha}(f)\,\partial^{\beta-\alpha}(g),$

where $\alpha$ is a multi-index.

## References

Title multi-index notation MultiindexNotation 2013-03-22 13:41:32 2013-03-22 13:41:32 matte (1858) matte (1858) 15 matte (1858) Definition msc 05-00 multi-index multi-indices