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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.
With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally: =: () ().. This formula can be used to derive a formula that computes the symbol of the composition of differential operators.
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
Multi-index notation; Multiplication; Musical notation; N. Nemeth Braille; Newman–Penrose formalism; Notation for differentiation; Notation in probability and ...
The essential ingredient of the proof is the following simple property, which uses multi-index notation for monomials in the variables X i. Lemma. The leading term of e λ t (X 1, ..., X n) is X λ. Proof.
Where denotes the partial derivative of order (see multi-index notation). When σ = 1 {\displaystyle \sigma =1} , G σ ( Ω ) {\displaystyle G^{\sigma }(\Omega )} coincides with the class of analytic functions C ω ( Ω ) {\displaystyle C^{\omega }(\Omega )} , but for σ > 1 {\displaystyle \sigma >1} there are compactly supported functions in ...