multi-index notation
Multi-indices form a powerful notational device for keeping track
of multiple derivatives
or multiple powers. In many respects
these resemble natural numbers
.
For example, one can define the factorial
, binomial coefficients
,
and derivatives for multi-indices.
Using these one can state traditional results such as the
multinomial theorem,
Leibniz’ rule, Taylor’s formula
, etc.
very concisely. In fact, the multi-dimensional results are more or
less obtained simply by replacing usual indices in with multi-indices.
See below for examples.
Definition
A multi-index is an -tuple
of non-negative integers . In other words,
. Usually, is the dimension of the underlying space.
Therefore, when dealing with multi-indices, is usually
assumed clear from the context.
Operations on multi-indices
For a multi-index , we define the length (or order) as
and the factorial as
If and
are two multi-indices,
their sum and difference is defined component-wise as
Thus . Also, if for all , then we write . For multi-indices , with , we define
For a point in (with standard coordinates) we define
Also, if is a smooth function, and
is a multi-index, we define
where are the standard unit vectors of .
Since is sufficiently smooth, the order in which the derivations are
performed is irrelevant. For multi-indices and , we thus
have
Examples
-
1.
If is a positive integer, and are complex numbers, the multinomial expansion states that
where and is a multi-index. (proof (http://planetmath.org/MultinomialTheoremProof))
-
2.
Leibniz’ rule: If are smooth functions, and is a multi-index, then
where is a multi-index.
References
-
1
M. Reed, B. Simon, Methods of Mathematical Physics,
I - Functional Analysis
, Academic Press, 1980.
Title | multi-index notation |
---|---|
Canonical name | MultiindexNotation |
Date of creation | 2013-03-22 13:41:32 |
Last modified on | 2013-03-22 13:41:32 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 15 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 05-00 |
Defines | multi-index |
Defines | multi-indices |