# Multi-index

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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 xα 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 Dα is used.