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{{Calculus|expanded=Multivariable calculus}}
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The [[mathematical notation]] of '''multi-indices''' simplifies formulae used in [[multivariable calculus]], [[partial differential equation]]s and the theory of [[distribution (mathematics)|distribution]]s, by generalising the concept of an integer [[index notation|index]] to an ordered [[tuple]]  of indices.  


==Multi-index notation==
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An ''n''-dimensional '''multi-index''' is an ''n''-[[tuple]]
 
:<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>
 
of [[natural number|non-negative integers]] (i.e. an element of the ''n''-[[dimension]]al [[set (mathematics)|set]] of [[natural number]]s, denoted <math>\mathbb{N}^n_0</math>).
 
For multi-indices <math>\alpha, \beta \in \mathbb{N}^n_0</math> and <math>x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n</math> one defines:
 
;Componentwise sum and difference
:<math>\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n)</math>
 
;[[Partial order]]
:<math>\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\{1,\ldots,n\}</math>
 
;Sum of components (absolute value)
:<math>| \alpha | = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>
 
;[[Factorial]]
:<math>\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n!</math>
 
;[[Binomial coefficient]]
:<math>\binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots\binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!}</math>
 
;[[Multinomial coefficient]]
:<math>\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} </math>
 
where <math>k:=|\alpha|\in\mathbb{N}_0\,\!</math>.
 
;[[Power (mathematics)|Power]]
:<math>x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \ldots x_n^{\alpha_n}</math>.
 
;Higher-order [[partial derivative]]
 
:<math>\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n}</math> 
where <math>\partial_i^{\alpha_i}:=\part^{\alpha_i} / \part x_i^{\alpha_i}</math> (see also [[4-gradient]]).
 
==Some applications==
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, <math>x,y,h\in\mathbb{C}^n</math> (or <math>\mathbb{R}^n</math>), <math>\alpha,\nu\in\mathbb{N}_0^n</math>, and <math>f,a_\alpha\colon\mathbb{C}^n\to\mathbb{C}</math> (or <math>\mathbb{R}^n\to\mathbb{R}</math>).
 
;[[Multinomial theorem]]
 
:<math> \biggl( \sum_{i=1}^n x_i\biggr)^k = \sum_{|\alpha|=k} \binom{k}{\alpha} \, x^\alpha</math>
 
;[[Multi-binomial theorem]]
 
:<math> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, x^\nu y^{\alpha - \nu}.</math>
 
Note that, since {{math|''x''+''y''}} is a vector and {{math|''α''}} is a multi-index, the expression on the left is short for {{math|(''x''<sub>1</sub>+''y''<sub>1</sub>)<sup>''α''<sub>1</sub></sup>...(''x''<sub>''n''</sub>+''y''<sub>''n''</sub>)<sup>''α''<sub>''n''</sub></sup>}}.
 
;[[Leibniz rule (generalized product rule)|Leibniz formula]]
For smooth functions ''f'' and ''g''
 
:<math>\partial^\alpha(fg) = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, \partial^{\nu}f\,\partial^{\alpha-\nu}g.</math>
 
;[[Taylor series]]
For an [[analytic function]] ''f'' in ''n'' variables one has
 
:<math>f(x+h) = \sum_{\alpha\in\mathbb{N}^n_0}^{}{\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}.</math>
In fact, for a smooth enough function, we have the similar '''Taylor expansion'''
:<math>f(x+h) = \sum_{|\alpha| \le n}{\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}+R_{n}(x,h),</math>
where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
:<math>R_n(x,h)= (n+1) \sum_{|\alpha| =n+1}\frac{h^\alpha}{\alpha !}\int_0^1(1-t)^n\partial^\alpha f(x+th)\,dt.</math>
 
;General [[partial differential operator]]
A formal ''N''-th order partial differential operator in ''n'' variables is written as
 
:<math>P(\partial) = \sum_{|\alpha| \le N}{}{a_{\alpha}(x)\partial^{\alpha}}.</math>
 
;[[Integration by parts]]
For smooth functions with [[compact support]] in a bounded domain <math>\Omega \subset \mathbb{R}^n</math> one has
 
:<math>\int_{\Omega}{}{u(\partial^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(\partial^{\alpha}u)v\,dx}.</math>
 
This formula is used for the definition of [[Distribution (mathematics)|distribution]]s and [[weak derivative]]s.
 
==An example theorem==
If <math>\alpha,\beta\in\mathbb{N}^n_0</math> are multi-indices and <math>x=(x_1,\ldots, x_n)</math>, then
 
:<math> \part^\alpha x^\beta =
\begin{cases}
\frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \hbox{if}\,\, \alpha\le\beta,\\
0 & \hbox{otherwise.} \end{cases}</math>
 
===Proof===
The proof follows from the [[power rule]] for the [[differential calculus|ordinary derivative]]; if ''&alpha;'' and ''&beta;'' are in {0,&nbsp;1,&nbsp;2,&nbsp;.&nbsp;.&nbsp;.}, then
 
:<math> \frac{d^\alpha}{dx^\alpha} x^\beta = \begin{cases} \frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \hbox{if}\,\, \alpha\le\beta, \\ 0 & \hbox{otherwise.} \end{cases}\qquad(1)</math>
 
Suppose <math>\alpha=(\alpha_1,\ldots, \alpha_n)</math>, <math>\beta=(\beta_1,\ldots, \beta_n)</math>, and <math>x=(x_1,\ldots, x_n)</math>. Then we have that
 
:<math>\begin{align}\part^\alpha x^\beta&= \frac{\part^{\vert\alpha\vert}}{\part x_1^{\alpha_1} \cdots \part x_n^{\alpha_n}} x_1^{\beta_1} \cdots x_n^{\beta_n}\\
&= \frac{\part^{\alpha_1}}{\part x_1^{\alpha_1}} x_1^{\beta_1} \cdots
\frac{\part^{\alpha_n}}{\part x_n^{\alpha_n}} x_n^{\beta_n}.\end{align}</math>
 
For each ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}, the function <math>x_i^{\beta_i}</math> only depends on <math>x_i</math>. In the above, each partial differentiation <math>\part/\part x_i</math> therefore reduces to the corresponding ordinary differentiation <math>d/dx_i</math>. Hence, from equation (1), it follows that <math>\part^\alpha x^\beta</math> vanishes if ''&alpha;<sub>i</sub>''&nbsp;>&nbsp;''&beta;<sub>i</sub>'' for at least one ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}. If this is not the case, i.e., if ''&alpha;''&nbsp;&le;&nbsp;''&beta;'' as multi-indices, then
 
:<math> \frac{d^{\alpha_i}}{dx_i^{\alpha_i}} x_i^{\beta_i} = \frac{\beta_i!}{(\beta_i-\alpha_i)!} x_i^{\beta_i-\alpha_i}</math>
for each <math>i</math> and the theorem follows. <math>\Box</math>
 
== See also ==
 
*[[Einstein notation]]
*[[Index notation]]
*[[Ricci calculus]]
 
== References ==
* Saint Raymond, Xavier (1991). ''Elementary Introduction to the Theory of Pseudodifferential Operators''. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9
 
{{PlanetMath attribution|id=4376|title=multi-index derivative of a power}}
 
[[Category:Combinatorics]]
[[Category:Mathematical notation]]
[[Category:Articles containing proofs]]

Latest revision as of 18:38, 8 January 2015

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