Multiindex notation
Part of a series of articles about  
Calculus  



Specialized 

Multiindex 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 ndimensional multiindex is an ntuple
of nonnegative integers (i.e. an element of the ndimensional set of natural numbers, denoted ).
For multiindices and one defines:
 Componentwise sum and difference
 Sum of components (absolute value)
where .
 Power
 .
 Higherorder partial derivative
where (see also 4gradient).
Some applications
The multiindex notation allows the extension of many formulae from elementary calculus to the corresponding multivariable case. Below are some examples. In all the following, (or ), , and (or ).
Note that, since x+y is a vector and α is a multiindex, the expression on the left is short for (x_{1}+y_{1})^{α1}...(x_{n}+y_{n})^{αn}.
For smooth functions f and g
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 (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
 General linear partial differential operator
A formal linear Nth order partial differential operator in n variables is written as
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 multiindices and , then
Proof
The proof follows from the power rule for the ordinary derivative; if α and β are in {0, 1, 2, . . .}, then
Suppose , , and . Then we have that
For each i in {1, . . ., n}, the function only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation (1), it follows that vanishes if α_{i} > β_{i} for at least one i in {1, . . ., n}. If this is not the case, i.e., if α ≤ β as multiindices, then
for each and the theorem follows.
See also
References
 Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0849371589
This article incorporates material from multiindex derivative of a power on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
