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.

Definition and basic properties

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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   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). Sometimes the notation   is also used.[1]

Some applications

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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   and  , 
Taylor series
For an analytic function   in   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  -th order partial differential operator in   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 theorem

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If   are multi-indices and  , then  

Proof

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The proof follows from the power rule for the ordinary derivative; if α and β are in  , then

  (1)

Suppose  ,  , and  . Then we have that  

For each   in  , 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   for at least one   in  . If this is not the case, i.e., if   as multi-indices, then   for each   and the theorem follows. Q.E.D.

See also

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References

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  1. ^ Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. p. 319. ISBN 0-12-585050-6.
  • Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9

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