Multi-index: Difference between revisions

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


Formally, multi-index <math>\alpha</math> is defined as
Formally, multi-index <math>\alpha</math> is defined as
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Basic definitions and notational conventions using multi-indices.
Basic definitions and notational conventions using multi-indices.


* the ''order'' or  ''length'' of <math>\alpha</math>
* The ''order'' or  ''length'' of <math>\alpha</math>
:<math>|\alpha| = \alpha_1+\alpha_2+\cdots+\alpha_n</math>
:<math>|\alpha| = \alpha_1+\alpha_2+\cdots+\alpha_n</math>
* factorial
* ''Factorial'' of a multi-index
:<math>\alpha ! = \alpha_1!\cdot\alpha_2!\cdots\alpha_n!</math>
:<math>\alpha ! = \alpha_1!\cdot\alpha_2!\cdots\alpha_n!</math>
 
* multidimensional power notation
[[Category:CZ Live]]
: If <math>x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n</math> and <math>\alpha=(\alpha_1,\,\alpha_1,\,\ldots,\alpha_n)</math> is a multi-index then <math>x^\alpha</math> is defined as
[[Category:Mathematics Workgroup]]
:<math>x^\alpha=(x_1^{\alpha_1}, x_2^{\alpha_2},\ldots,x_n^{\alpha_n})</math>
[[Category:Stub Articles]]
* The following notation is used for partial derivatives of a function <math>f: \mathbb{R}^n\mapsto \mathbb{R}</math>
:<math> D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots \partial x_n^{\alpha_n}}</math>
:Remark: sometimes the symbol <math>\partial^\alpha</math> instead of <math>D^\alpha</math> is used.

Latest revision as of 04:55, 26 December 2007

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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.