imported>Abhiramakella: /* Proof */

2023-09-10T20:57:58Z

Proof

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{{Short description|Mathematical notation}}{{Calculus|expanded=Multivariable calculus}}
'''Multi-index notation''' is a [[mathematical notation]] that simplifies formulas 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.

==Definition and basic properties==

An ''n''-dimensional '''multi-index''' is an <math display="inline">n</math>-[[tuple]]

:<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>

of [[non-negative integer]]s (i.e. an element of the ''<math display="inline">n</math>''-[[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 display="block">\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 display="block">\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n},</math> where <math>\partial_i^{\alpha_i}:=\partial^{\alpha_i} / \partial x_i^{\alpha_i}</math> (see also [[4-gradient]]). Sometimes the notation <math>D^{\alpha} = \partial^{\alpha}</math> is also used.<ref>{{cite book |first=M. |last=Reed |first2=B. |last2=Simon |title=Methods of Modern Mathematical Physics: Functional Analysis I |edition=Revised and enlarged |publisher=Academic Press |location=San Diego |year=1980 |isbn=0-12-585050-6| page=319 }}</ref>

==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\Complex^n</math> (or <math>\R^n</math>), <math>\alpha,\nu\in\N_0^n</math>, and <math>f,g,a_\alpha\colon\Complex^n\to\Complex</math> (or <math>\R^n\to\R</math>).

;[[Multinomial theorem]]
:<math> \left( \sum_{i=1}^n x_i\right)^k = \sum_{|\alpha|=k} \binom{k}{\alpha} \, x^\alpha</math>
;[[Multi-binomial theorem]]
:<math display="block"> (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''1 + ''y''1)''α''1⋯(''x''''n'' + ''y''''n'')''α''''n''}}.
;[[Leibniz rule (generalized product rule)|Leibniz formula]]
:For smooth functions <math display="inline">f</math> and <math display="inline">g</math>,<math display="block">\partial^\alpha(fg) = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, \partial^{\nu}f\,\partial^{\alpha-\nu}g.</math>
;[[Taylor series]]
:For an [[analytic function]] <math display="inline">f</math> in ''<math display="inline">n</math>'' variables one has <math display="block">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 display="block">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 display="block">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 linear [[partial differential operator]]
:A formal linear <math display="inline">N</math>-th order partial differential operator in <math display="inline">n</math> variables is written as <math display="block">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 \R^n</math> one has <math display="block">\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 display="block"> \partial^\alpha x^\beta = \begin{cases}
\frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \text{if}~ \alpha\le\beta,\\
0 & \text{otherwise.}
\end{cases}</math>

===Proof===
The proof follows from the [[power rule]] for the [[differential calculus|ordinary derivative]]; if ''α'' and ''β'' are in <math display="inline">\{0, 1, 2,\ldots\}</math>, then
{{NumBlk||<math display="block"> \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}</math>|{{EquationRef|1}}}}

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 display="block">\begin{align}\partial^\alpha x^\beta&= \frac{\partial^{\vert\alpha\vert}}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} x_1^{\beta_1} \cdots x_n^{\beta_n}\\
&= \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}} x_1^{\beta_1} \cdots
\frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}} x_n^{\beta_n}.\end{align}</math>

For each <math display="inline">i</math> in <math display="inline">\{ 1, \ldots , n\}</math>, the function <math>x_i^{\beta_i}</math> only depends on <math>x_i</math>. In the above, each partial differentiation <math>\partial/\partial x_i</math> therefore reduces to the corresponding ordinary differentiation <math>d/dx_i</math>. Hence, from equation ({{EquationNote|1}}), it follows that <math>\partial^\alpha x^\beta</math> vanishes if <math display="inline">\alpha_i > \beta_i</math> for at least one <math display="inline">i</math> in <math display="inline">\{ 1, \ldots , n\}</math>. If this is not the case, i.e., if <math display="inline">\alpha \leq \beta</math> as multi-indices, then
<math display="block"> \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. [[Q.E.D.]]

== See also ==

*[[Einstein notation]]
*[[Index notation]]
*[[Ricci calculus]]

== References ==
{{Reflist}}
* 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}}

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[[Category:Combinatorics]]
[[Category:Mathematical notation]]
[[Category:Articles containing proofs]]

Multi-index notation - Revision history

imported>Abhiramakella: /* Proof */