Multi-index

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A checked version of this page, approved on 16 November 2011, was based off this revision.

In mathematics, multi-index is an n-tuple of non-negative integers. Multi-indices are widely used in multivariable calculus to denote e.g. partial derivatives and multidimensional power function. Many formulas known from the univariable calculus (i.e. over the real line) carry on to $\mathbb{R}^n$ by simple replacing usual indices with multi-indices.

Formally, multi-index $α$ is defined as

$\alpha=(\alpha_1,\,\alpha_1,\,\ldots,\alpha_n)$ , where $\alpha_i\in\mathbb{N}\cup\{0\}.$

Basic definitions and notational conventions using multi-indices.

The order or length of $α$

$|\alpha| = \alpha_1+\alpha_2+\cdots+\alpha_n$

Factorial of a multi-index is the product

$\alpha ! = \alpha_1! \alpha_2! \cdots \alpha_n!$

multidimensional power notation

If $x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ and $\alpha=(\alpha_1,\,\alpha_1,\,\ldots,\alpha_n)$ is a multi-index then

x α

is defined as the product

$x^\alpha=x_1^{\alpha_1} x_2^{\alpha_2} \ldots x_n^{\alpha_n}$

The following notation is used for partial derivatives of a function $f: \mathbb{R}^n\mapsto \mathbb{R}$

$D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots \partial x_n^{\alpha_n}}$

Remark: sometimes the symbol $\partial^\alpha$ instead of

D α

is used.

Some content on this page may previously have appeared on Citizendium.

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Category: Mathematics

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