Multi-index notation

The mathematical notation of multi-indices simplifies formulae 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.

Multi-index notation

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

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

of non-negative integers. 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 $|\alpha|=k\,$

• 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 $\partial_i^{\alpha_i}:=\part^{\alpha_i} / \part x_i^{\alpha_i}$

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 $f,a_\alpha\colon\mathbb{C}^n\to\mathbb{C}$ (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

$(x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, x^\nu y^{\alpha - \nu}.$

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 partial differential operator

A formal 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

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