Multi-index notation

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ==Definition and basic properties== An n-dimensional multi-index is an n -tuple \alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of non-negative integers (i.e. an element of the n -dimensional set of natural numbers, denoted \mathbb{N}^n_0 ). 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 \binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots\binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!} Multinomial coefficient \binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} where k:=|\alpha|\in\mathbb{N}_0 . 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}:=\partial^{\alpha_i} / \partial x_i^{\alpha_i} (see also 4-gradient). Sometimes the notation D^{\alpha} = \partial^{\alpha} is also used. ==Some applications== The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, x,y,h\in\Complex^n (or \R^n ), and \alpha,\beta, \nu\in\N_0^n . ===Derivative of a monomial=== \partial^\alpha x^\beta = \begin{cases} \frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha}, & \text{if}~ \alpha\le\beta,\\ 0, & \text{otherwise.} \end{cases} ===Multinomial theorem=== \left( \sum_{i=1}^n x_i\right)^k = \sum_ \int_{\R^n} {(\partial^{\alpha}u)v\,dx}. This formula is used for the definition of weak derivatives of distributions.

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