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

<math>\alpha = (\alpha_{1}, \alpha_{2},\ldots,\alpha_{n})</math>

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

<math>\alpha \pm \beta:= (\alpha_{1} \pm \beta_{1},\,\alpha_{2} \pm \beta_{2}, \ldots, \,\alpha_{n} \pm \beta_{n})</math>

<math>\alpha \le \beta \quad \Leftrightarrow \quad \alpha_{i} \le \beta_{i} \quad \forall\,i</math>

<math>| \alpha | = \alpha_{1} + \alpha_{2} + \ldots + \alpha_{n}</math>

<math>\alpha ! = \alpha_{1}! \alpha_{2}! \ldots \alpha_{n}!</math>

<math>{\alpha \choose \beta} = \frac{\alpha!}{(\alpha - \beta)! \, \beta!}={\alpha_{1} \choose \beta_{1}}{\alpha_{2} \choose \beta_{2}}\ldots{\alpha_{n} \choose \beta_{n}}</math>

<math>\mathbf{x}^\alpha = x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \ldots x_{n}^{\alpha_{n}}</math>

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

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:

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

Leibniz formula: for smooth functions u, v

<math>D^{\alpha}(uv) = \sum_{\nu \le \alpha}^{}{{\alpha \choose \nu}D^{\nu}u\,D^{\alpha-\nu}v}</math>

Taylor series: for an analytic function f one has

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

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

<math>P(D) = \sum_{|\alpha| \le N}{}{a_{\alpha}(x)D^{\alpha}}</math>

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

<math>\int_{\Omega}{}{u(D^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(D^{\alpha}u)v\,dx}</math>

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

Theorem

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

<math> \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.</math>

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

<math> \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.</math>. (1)

Suppose <math>i=(i_1,\ldots, i_n)</math>, <math>k=(k_1,\ldots, k_n)</math>, and <math>x=(x_1,\ldots, x_n)</math>. Then we have that <math> \part^i x^k</math> <math> =</math> <math> \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}</math>

<math> =</math> <math> \frac{\part^{i_1}}{\part x_1^{i_1}} x_1^{k_1} \cdot \cdots \cdot

\frac{\part^{i_n}}{\part x_n^{i_n}} x_n^{k_n}</math>.

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