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.

An n-dimensional multi-index is an n-tuple

of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted ${\displaystyle \mathbb {N} _{0}^{n}}$).

For multi-indices ${\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}}$ and ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}}$ one defines:

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