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Multi-index notation

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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.

Contents

Definition and basic propertiesEdit

An n-dimensional multi-index is an n-tuple

 

of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted  ).

For multi-indices   and   one defines:

Componentwise sum and difference
 
Partial order
 
Sum of components (absolute value)
 
Factorial
 
Binomial coefficient
 
Multinomial coefficient
 

where  .

Power
 .
Higher-order partial derivative
 

where   (see also 4-gradient).

Some applicationsEdit

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,   (or  ),  , and   (or  ).

Multinomial theorem
 
Multi-binomial theorem
 

Note that, since x+y is a vector and α is a multi-index, the expression on the left is short for (x1+y1)α1...(xn+yn)αn.

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 (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 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 theoremEdit

If   are multi-indices and  , then

 

ProofEdit

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 αi > βi for at least one i in {1, . . ., n}. If this is not the case, i.e., if α ≤ β as multi-indices, then

 

for each   and the theorem follows.  

See alsoEdit

ReferencesEdit

  • Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9

This article incorporates material from multi-index derivative of a power on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.