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.

Definition and basic properties

An n-dimensional multi-index is an -tuple

of non-negative integers (i.e. an element of the -dimensional set of natural numbers, denoted ).

For multi-indices and , one defines:

Componentwise sum and difference

Partial order

Sum of components (absolute value)

Factorial

Binomial coefficient

Multinomial coefficient where . Power . Higher-order partial derivative where (see also 4-gradient). Sometimes the notation 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, (or ), and .

Derivative of a monomial

Multinomial theorem

Multi-binomial theorem

Note that, since is a vector and is a multi-index, the expression on the left is short for . Leibniz formula

For smooth functions and , Taylor series

For an analytic function in variables one has In fact, for a smooth enough function, we have the similar Taylor expansion where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets Integration by parts

For smooth functions and with compact support in , This formula is used for the definition of weak derivatives of distributions.

Multi-index notation

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