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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.