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
- 1 M. Reed, B. Simon, Methods of Mathematical Physics, I - Functional Analysis, Academic Press, 1980.
|Date of creation||2013-03-22 13:41:32|
|Last modified on||2013-03-22 13:41:32|
|Last modified by||matte (1858)|