Multi-index notation

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The notion of multi-indices simplifies formulae used in the multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an array of indices.

An n-dimensional multi-index is a vector

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

with integers $α i$ . For multi-indices $\alpha, \beta \in \mathbb{N}^n$ and $\mathbf{x} = (x_{1}, x_{2}, \ldots, x_{n}) \in \mathbb{R}^n$ one defines:

$\alpha \pm \beta:= (\alpha_{1} \pm \beta_{1},\,\alpha_{2} \pm \beta_{2}, \ldots, \,\alpha_{n} \pm \beta_{n})$

$\alpha \le \beta \quad \Leftrightarrow \quad \alpha_{i} \le \beta_{i} \quad \forall\,i$

$| \alpha | = \alpha_{1} + \alpha_{2} + \ldots + \alpha_{n}$

$\alpha ! = \alpha_{1}! \alpha_{2}! \ldots \alpha_{n}!$

${\alpha \choose \beta} = \frac{\alpha!}{(\alpha - \beta)! \, \beta!}={\alpha_{1} \choose \beta_{1}}{\alpha_{2} \choose \beta_{2}}\ldots{\alpha_{n} \choose \beta_{n}}$

$\mathbf{x}^\alpha = x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \ldots x_{n}^{\alpha_{n}}$

$D^{\alpha} := D_{1}^{\alpha_{1}} D_{2}^{\alpha_{2}} \ldots D_{n}^{\alpha_{n}}$ where $D_{i}^{j}:=\part^{j} / \part x_{i}^{j}$

The notation allows to extend many formula from elementary calculus to the corresponding multi-variable case. Some examples of common applications of multi-index notations:

Multinomial expansion:

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

Leibniz formula: for smooth functions u, v

$D^{\alpha}(uv) = \sum_{\nu \le \alpha}^{}{{\alpha \choose \nu}D^{\nu}u\,D^{\alpha-\nu}v}$

Taylor series: for an analytic function f one has

$f(\mathbf{x}+\mathbf{h}) = \sum_{|\alpha| \ge 0}^{}{\frac{D^{\alpha}f(\mathbf{x})}{\alpha !}\mathbf{h}^{\alpha}}$

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

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

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

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

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

[edit] Theorem

Theorem If $i, k$ are multi-indices in $\mathbb{N}^n$ , and $x=(x_1,\ldots, x_n)$ , then

$\part^i x^k = \left\{\begin{matrix} \frac{k!}{(k-i)!} x^{k-i} & \hbox{if}\,\, i\le k\\ 0 & \hbox{otherwise.} \end{matrix}\right.$

Proof. The proof follows from the corresponding rule for the ordinary derivative; if $i, k$ are in $0,1,2,\ldots$ , then

$\frac{d^i}{dx^i} x^k = \left\{ \begin{matrix} \frac{k!}{(k-i)!} x^{k-i} & \hbox{if}\,\, i\le k, \\ 0 & \hbox{otherwise.} \end{matrix}\right.$ . (1)

Suppose $i=(i_1,\ldots, i_n)$ , $k=(k_1,\ldots, k_n)$ , and $x=(x_1,\ldots, x_n)$ . Then we have that

$\part^i x^k$

=

$\frac{\part^{\vert i\vert}}{\part x_1^{i_1} \cdots \part x_n^{i_n}} x_1^{k_1} \cdots x_n^{k_n}$

=

$\frac{\part^{i_1}}{\part x_1^{i_1}} x_1^{k_1} \cdots \frac{\part^{i_n}}{\part x_n^{i_n}} x_n^{k_n}$ .

For each $r=1,\ldots, n$ , the function $x_r^{k_r}$ only depends on $x r$ . In the above, each partial differentiation $\part/\part x_r$ therefore reduces to the corresponding ordinary differentiation $d / d x r$ . Hence, from equation 1, it follows that $\part^i x^k$ vanishes if $i r > k r$ for any $r=1,\ldots, n$ . If this is not the case, i.e., if $i\le k$ as multi-indices, then for each $r$ ,