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The notion of multi-indices simplifies formula used in the calculus of several variables, 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}:=\partial^{j} / \partial 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 analytical 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 instrumental for the definition of distributions and weak derivatives.
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