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 $\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$ of non-negative 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 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

$$|\alpha |={\alpha }_1+\mathrm{\cdots }+{\alpha }_n,$$

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 multi-indices, their sum and difference is defined component-wise 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 multi-indices $\alpha ,\beta$, 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 ${ℝ}^n$ (with standard coordinates) we define

$$x^{\alpha }=\prod _{k=1}^n{x}_k^{{\alpha }_k}.$$

Also, if $f:{ℝ}^n\to ℝ$ is a smooth function, and $\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$ is a multi-index, 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 ${ℝ}^n$. 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. 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 multi-index. (proof (http://planetmath.org/MultinomialTheoremProof))

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

Title	multi-index notation
Canonical name	MultiindexNotation
Date of creation	2013-03-22 13:41:32
Last modified on	2013-03-22 13:41:32
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	15
Author	matte (1858)
Entry type	Definition
Classification	msc 05-00
Defines	multi-index
Defines	multi-indices