multi-index notation

Multi-indices form a powerful notational device for keeping trackof multiple derivatives or multiple powers. In many respectsthese resemble natural numbers.For example, one can define the factorial, binomial coefficients,and derivatives for multi-indices.Using these one can state traditional results such as themultinomial theorem,Leibniz’ rule, Taylor’s formula, etc.very concisely. In fact, the multi-dimensional results are more orless obtained simply by replacing usual indices in $\mathbb{N}$ with multi-indices.See below for examples.

DefinitionA multi-index is an $n$-tuple$\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$ of non-negative integers ${\alpha }_1,\mathrm{\dots },{\alpha }_n$. In other words,$\alpha \in {\mathbb{N}}^n$. Usually, $n$ is the dimension of the underlying space.Therefore, when dealing with multi-indices, $n$ is usuallyassumed clear from the context.

Operations on multi-indices

For a multi-index $\alpha$, we define the length (or order) as

$$|\alpha |={\alpha }_1+\mathrm{\cdots }+{\alpha }_n,$$

and the factorial as

$$\alpha !=\prod _{k=1}^n{\alpha }_k!.$$

If $\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$ and$\beta =({\beta }_1,\mathrm{\dots },{\beta }_n)$ are two multi-indices,their sum and difference is defined component-wise as

$\alpha +\beta$$=$$({\alpha }_1+{\beta }_1,\mathrm{\dots },{\alpha }_n+{\beta }_n),$

$\alpha -\beta$$=$$({\alpha }_1-{\beta }_1,\mathrm{\dots },{\alpha }_n-{\beta }_n).$

Thus $|\alpha \pm \beta |=|\alpha |\pm |\beta |$.Also, if ${\beta }_k\le {\alpha }_k$ for all $k=1,\mathrm{\dots },n$, then we write$\beta \le \alpha$. For multi-indices $\alpha ,\beta$, with$\beta \le \alpha$, we define

$$\left(\genfrac{}{}{0pt}{}{\alpha }{\beta }\right)=\frac{\alpha !}{(\alpha -\beta )!\beta !}.$$

For a point $x=(x_1,\mathrm{\dots },x_n)$ in ${ℝ}^n$ (withstandard coordinates) we define

$$x^{\alpha }=\prod _{k=1}^n{x}_k^{{\alpha }_k}.$$

Also, if $f:{ℝ}^n\to ℝ$ is a smooth function, and$\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$ is a multi-index, we define

$${\partial }^{\alpha }f=\frac{{\partial }^{|\alpha |}}{{\partial }^{{\alpha }_1}{e}_1\mathrm{\cdots }{\partial }^{{\alpha }_n}{e}_n}f,$$

where $e_1,\mathrm{\dots },e_n$ are the standard unit vectors of ${ℝ}^n$.Since $f$ is sufficiently smooth, the order in which the derivations areperformed is irrelevant. For multi-indices $\alpha$ and $\beta$, we thushave

$${\partial }^{\alpha }{\partial }^{\beta }={\partial }^{\alpha +\beta }={\partial }^{\beta +\alpha }={\partial }^{\beta }{\partial }^{\alpha }.$$

Examples

1. If $n$ is a positive integer, and $x_1,\mathrm{\dots },x_k$ arecomplex numbers, the multinomial expansion states that

   $${(x_1+\mathrm{\cdots }+x_k)}^n=n!\sum _{|\alpha |=n}\frac{x^{\alpha }}{\alpha !},$$

   where $x=(x_1,\mathrm{\dots },x_k)$ and $\alpha$ is a multi-index.(proof (http://planetmath.org/MultinomialTheoremProof))

2. Leibniz’ rule: If $f,g:{ℝ}^n\to ℝ$ are smooth functions, and $\beta$ isa multi-index, then

   $${\partial }^{\beta }(fg)=\sum _{\alpha \le \beta }\left(\genfrac{}{}{0pt}{}{\beta }{\alpha }\right){\partial }^{\alpha }(f){\partial }^{\beta -\alpha }(g),$$

   where $\alpha$ is a multi-index.