# 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 analysis to denote e.g. partial derivatives and multidimensional power function. Many formulas known from the one dimension one (i.e. the real line) carry on to by simple replacing usual indices with multi-indices.

Formally, multi-index is defined as

- , where

Basic definitions and notational conventions using multi-indices.

- The
*order*or*length*of

*Factorial*of a multi-index

- multidimensional power notation

- If and is a multi-index then is defined as

- The following notation is used for partial derivatives of a function

- Remark: sometimes the symbol instead of is used.