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 n-tuple
of non-negative integers.
For multi-indices and one defines:
;Componentwise sum and difference
;Partial order
;Sum of components
;Factorial
;Binomial coefficient
;Multinomial coefficient
where.
;Power
;Higher-order partial derivative
where .

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, ,, and .
;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 f and g
;Taylor series
For an analytic function f in n variables one has
In fact, for a smooth enough function, we have the similar Taylor expansion
where the last term depends on the exact version of Taylor's formula. For instance, for the Cauchy formula, one gets
;General linear partial differential operator
A formal linear N-th order partial differential operator in n variables is written as
;Integration by parts
For smooth functions with compact support in a bounded domain one has
This formula is used for the definition of distributions and weak derivatives.

An example theorem

If are multi-indices and, then

Proof

The proof follows from the power rule for the ordinary derivative; if α and β are in, then
Suppose,, and. Then we have that
For each i in, the function only depends on. In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation. Hence, from equation, it follows that vanishes if αi > βi for at least one i in. If this is not the case, i.e., if α ≤ β as multi-indices, then
for each and the theorem follows.