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.
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
α , β ∈
, one defines: The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case.
Below are some examples.
α , ν ∈
α , β ∈
are multi-indices and
β
β !
( β − α ) !
β − α
α ≤ β ,
{\displaystyle \partial ^{\alpha }x^{\beta }={\begin{cases}{\frac {\beta !
}}\end{cases}}}
The proof follows from the power rule for the ordinary derivative; if α and β are in
, then Suppose
β = (
{\displaystyle {\begin{aligned}\partial ^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}}
In the above, each partial differentiation
therefore reduces to the corresponding ordinary differentiation
Hence, from equation (1), it follows that
If this is not the case, i.e., if
α ≤ β
as multi-indices, then
and the theorem follows.
This article incorporates material from multi-index derivative of a power on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.