Multi-index notation

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Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

Definition and basic properties#edit

An n-dimensional multi-index is an ${\textstyle n}$ -tuple

${\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}$

of non-negative integers (i.e. an element of the ${\textstyle n}$ -dimensional set of natural numbers, denoted ${\displaystyle \mathbb {N} _{0}^{n$ ).

For multi-indices ${\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n$ and ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n$ , one defines:

Componentwise sum: ${\displaystyle \alpha +\beta =(\alpha _{1}+\beta _{1},\,\alpha _{2}+\beta _{2},\ldots ,\,\alpha _{n}+\beta _{n})}$
Partial order: ${\displaystyle \alpha \leq \beta \quad \Leftrightarrow \quad \alpha _{i}\leq \beta _{i}\quad \forall \,i\in \{1,\ldots ,n\$
Difference: If ${\displaystyle \alpha \leq \beta }$ then ${\displaystyle (\beta -\alpha )_{i}=\beta _{i}-\alpha _{i$; else the difference is undefined.
Length: ${\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n$
Factorial: ${\displaystyle \alpha !=\alpha _{1}!\cdot \alpha _{2}!\cdots \alpha _{n}!}$
Binomial coefficient: ${\displaystyle {\binom {\alpha }{\beta ={\binom {\alpha _{1{\beta _{1{\binom {\alpha _{2{\beta _{2\cdots {\binom {\alpha _{n{\beta _{n={\frac {\alpha !}{\beta !(\alpha -\be$
Multinomial coefficient: ${\displaystyle {\binom {k}{\alpha ={\frac {k!}{\alpha _{1}!\alpha _{2}!\cdots \alpha _{n}!={\frac {k!}{\alpha !$ where ${\displaystyle k:=|\alpha |\in \mathbb {N} _{0$ .
Power: ${\displaystyle x^{\alpha }=x_{1}^{\alpha _{1x_{2}^{\alpha _{2\ldots x_{n}^{\alpha _{n$ .
Higher-order partial derivative: ${\displaystyle \partial ^{\alpha }=\partial _{1}^{\alpha _{1\partial _{2}^{\alpha _{2\ldots \partial _{n}^{\alpha _{n,}$

where ${\displaystyle \partial _{i}^{\alpha _{i:=(\partial /\partial x_{i})^{\alpha _{i.}$ ^[1]

Some applications#edit

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, ${\displaystyle x,y,h\in \mathbb {C} ^{n$ (or ${\displaystyle \mathbb {R} ^{n$ ), and ${\displaystyle \alpha ,\beta ,\nu \in \mathbb {N} _{0}^{n$ .

Derivative of a monomial#edit

{{mathimg|f94325797acf5f2a5f4d6ed8917870c23b298f0d|width=34.727ex|height=7.509ex|va=-3.171ex|alt={\displaystyle \partial ^{\alpha }x^{\beta }={\begin{cases}{\frac {\beta !}{(\beta -\alpha )!x^{\beta -\alpha },&{\text{if~\alpha \leq \beta ,\\0,&{\text{otherwise.\end{cases}}

Multinomial theorem#edit

${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)^{k}=\sum _{|\alpha |=k}{\binom {k}{\alpha \,x^{\alpha }.}$

Multi-binomial theorem#edit

{{mathimg|db39d7304e8fba5771d4a36ee307044d1b1df7b4|width=28.23ex|height=6.843ex|va=-3.171ex|alt={\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu \,x^{\nu }y^{\alpha -\nu }.}}} Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x₁ + y₁)^α₁⋯(x_n + y_n)^α_n.

Leibniz formula#edit

For smooth functions ${\textstyle f}$ and ${\textstyle g}$ , ${\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu \,\partial ^{\nu }f\,\partial ^{\alpha -\nu }g.}$

Taylor series#edit

For an analytic function ${\textstyle f}$ in ${\textstyle n}$ variables one has ${\displaystyle f(x+h)=\sum _{\alpha \in \mathbb {N} _{0}^{n\frac {\partial ^{\alpha }f(x)}{\alpha !h^{\alpha .}$ In fact, for a smooth enough function, we have the similar Taylor expansion ${\displaystyle f(x+h)=\sum _{|\alpha |\leq n}\frac {\partial ^{\alpha }f(x)}{\alpha !h^{\alpha +R_{n}(x,h),}$ 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 ${\displaystyle R_{n}(x,h)=(n+1)\sum _{|\alpha |=n+1}{\frac {h^{\alpha {\alpha !\int _{0}^{1}(1-t)^{n}\partial ^{\alpha }f(x+th)\,dt.}$

Integration by parts#edit

For smooth functions ${\displaystyle u}$ and ${\displaystyle v}$ with compact support in ${\displaystyle \mathbb {R} ^{n$ , ${\displaystyle \int _{\mathbb {R} ^{nu(\partial ^{\alpha }v)\,dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n{(\partial ^{\alpha }u)v\,dx}.}$ This formula is used for the definition of weak derivatives of distributions.

References#edit

^ Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. p. 319. ISBN 0-12-585050-6.

Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9

This article incorporates material from multi-index derivative of a power on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Categories Combinatorics Mathematical notation

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