multi-index notation

Multi-indices form a powerful notational device for keeping track of multipleMathworldPlanetmath derivativesPlanetmathPlanetmath or multiple powers. In many respects these resemble natural numbersMathworldPlanetmath. For example, one can define the factorialMathworldPlanetmath, binomial coefficientsMathworldPlanetmath, and derivatives for multi-indices. Using these one can state traditional results such as the multinomial theorem, Leibniz’ rule, Taylor’s formulaMathworldPlanetmathPlanetmath, etc. very concisely. In fact, the multi-dimensional results are more or less obtained simply by replacing usual indices in with multi-indices. See below for examples.

Definition A multi-index is an n-tuple α=(α1,,αn) of non-negative integers α1,,αn. In other words, αn. Usually, n is the dimensionMathworldPlanetmathPlanetmath 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 α, we define the length (or order) as


and the factorial as


If α=(α1,,αn) and β=(β1,,βn) are two multi-indices, their sum and differencePlanetmathPlanetmath is defined component-wise as

α+β = (α1+β1,,αn+βn),
α-β = (α1-β1,,αn-βn).

Thus |α±β|=|α|±|β|. Also, if βkαk for all k=1,,n, then we write βα. For multi-indices α,β, with βα, we define


For a point x=(x1,,xn) in n (with standard coordinates) we define


Also, if f:n is a smooth functionMathworldPlanetmath, and α=(α1,,αn) is a multi-index, we define


where e1,,en are the standard unit vectors of n. Since f is sufficiently smooth, the order in which the derivationsMathworldPlanetmath are performed is irrelevant. For multi-indices α and β, we thus have



  1. 1.

    If n is a positive integer, and x1,,xk are complex numbers, the multinomial expansion states that


    where x=(x1,,xk) and α is a multi-index. (proof (

  2. 2.

    Leibniz’ rule: If f,g:n are smooth functions, and β is a multi-index, then


    where α is a multi-index.


  • 1 M. Reed, B. Simon, Methods of Mathematical Physics, I - Functional AnalysisMathworldPlanetmath, Academic Press, 1980.
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