Multi-index notation

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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 a vector of indices.

[edit] Multi-index notation

An n-dimensional multi-index is a vector

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

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

${\alpha \choose \beta} ={\alpha_1 \choose \beta_1}{\alpha_2 \choose \beta_2}\cdots{\alpha_n \choose \beta_n}$

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

[edit] Some applications

The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Here are some examples:

[edit] Multinomial theorem

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

[edit] Leibniz formula

For smooth functions f and g

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

[edit] 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

$R_n(x,h)= (n+1) \sum_{|\alpha| =n+1}\frac{h^\alpha}{\alpha !}\int_0^1(1-t)^n\partial^\alpha f(x+th)\,dt.$

[edit] General partial differential operator

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

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

[edit] 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.

[edit] An example theorem

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

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

[edit] 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 $d / d x i$ . 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