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[edit]

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

Some applications[edit]

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 theorem[edit]

If are multi-indices and , then

Proof[edit]

The proof follows from the power rule for the ordinary derivative; if α and β are in {0, 1, 2, …}, then

 

 

 

 

(1)

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. Q.E.D.

See also[edit]

References[edit]

  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

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