Multi-index notation

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

An n-dimensional multi-index is an n-tuple

\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)

of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted $\mathbb{N}^n_0$ ).

For multi-indices $\alpha, \beta \in \mathbb{N}^n_0$ and $x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$ one defines:

Componentwise sum and difference: $\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n)$

Partial order: $\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\{1,\ldots,n\}$

Sum of components (absolute value): $| \alpha | = \alpha_1 + \alpha_2 + \cdots + \alpha_n$

Factorial: $\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n!$

Binomial coefficient: $\binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots\binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!}$

Multinomial coefficient: $\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!}$

where $k:=|\alpha |\in \mathbb {N} _{0}$ .

Power: $x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \ldots x_n^{\alpha_n}$ .

Higher-order partial derivative

\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n}

where $\partial_i^{\alpha_i}:=\part^{\alpha_i} / \part x_i^{\alpha_i}$ (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, $x,y,h\in\mathbb{C}^n$ (or $\mathbb {R} ^{n}$ ), $\alpha,\nu\in\mathbb{N}_0^n$ , and $f,g,a_{\alpha }\colon {\mathbb {C}}^{n}\to {\mathbb {C}}$ (or $\mathbb{R}^n\to\mathbb{R}$ ).

Multinomial theorem

\biggl( \sum_{i=1}^n x_i\biggr)^k = \sum_{|\alpha|=k} \binom{k}{\alpha} \, x^\alpha

Multi-binomial theorem

(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 $(x 1 + y 1) α 1 ...(x n + y n) α n$ .

Leibniz formula

For smooth functions f and g

\partial^\alpha(fg) = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, \partial^{\nu}f\,\partial^{\alpha-\nu}g.

Taylor series

For an analytic function f in n variables one has

f(x+h) = \sum_{\alpha\in\mathbb{N}^n_0}^{}{\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}.

In fact, for a smooth enough function, we have the similar Taylor expansion

f(x+h) = \sum_{|\alpha| \le 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

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.

General linear partial differential operator

A formal linear N-th order partial differential operator in n variables is written as

P(\partial) = \sum_{|\alpha| \le N}{}{a_{\alpha}(x)\partial^{\alpha}}.

Integration by parts

For smooth functions with compact support in a bounded domain $\Omega \subset \mathbb{R}^n$ one has

\int_{\Omega}{}{u(\partial^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(\partial^{\alpha}u)v\,dx}.

This formula is used for the definition of distributions and weak derivatives.

An example theorem

If $\alpha,\beta\in\mathbb{N}^n_0$ are multi-indices and $x=(x_1,\ldots, x_n)$ , then

\part^\alpha x^\beta = \begin{cases} \frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \hbox{if}\,\, \alpha\le\beta,\\ 0 & \hbox{otherwise.} \end{cases}

Proof

The proof follows from the power rule for the ordinary derivative; if α and β are in {0, 1, 2, . . .}, then

\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)

Suppose $\alpha=(\alpha_1,\ldots, \alpha_n)$ , $\beta=(\beta_1,\ldots, \beta_n)$ , and $x=(x_1,\ldots, x_n)$ . Then we have that

\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}

For each i in {1, . . ., n}, the function $x_i^{\beta_i}$ only depends on $x_{i}$ . In the above, each partial differentiation $\part/\part x_i$ therefore reduces to the corresponding ordinary differentiation $d/dx_i$ . Hence, from equation (1), it follows that $\part^\alpha x^\beta$ vanishes if α_i > β_i for at least one i in {1, . . ., n}. If this is not the case, i.e., if α ≤ β as multi-indices, then

\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}

for each $i$ and the theorem follows. $\Box$

References

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

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