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}:=\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 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.