multi-index notation

multi-index notation

Multi-indices

form a powerful notational device for keeping track
of

multiple

derivatives

or multiple powers. In many respects
these resemble

natural numbers

.
For example, one can define the

factorial

,

binomial coefficients

,
and derivatives for multi-indices.
Using these one can state traditional results such as the

multinomial theorem

,
Leibniz’ rule, Taylor’s

formula

, etc.
very concisely. In fact, the multi-dimensional results are more or
less obtained simply by replacing usual indices in

ℕ

with multi-indices.
See below for examples.

Definition

A

multi-index

is an

n

-tuple

α

=

(

α

1

,

…

,

α

n

)

of non-negative

integers

α

1

,

…

,

α

n

. In other words,

α

∈

ℕ

n

. Usually,

n

is the

dimension

of the underlying space.
Therefore, when dealing with multi-indices,

n

is usually
assumed clear from the

context

.

Operations on multi-indices

For a multi-index

α

, we define the

length

(or

order

) as

|

α

|

=

α

1

+

⋯

+

α

n

,

and the

factorial

as

α

!

=

∏

k

=

1

n

α

k

!

.

If

α

=

(

α

1

,

…

,

α

n

)

and

β

=

(

β

1

,

…

,

β

n

)

are two multi-indices,
their sum and

difference

is defined component-wise as

α

+

β

=

(

α

1

+

β

1

,

…

,

α

n

+

β

n

)

,

α

-

β

=

(

α

1

-

β

1

,

…

,

α

n

-

β

n

)

.

Thus

|

α

±

β

|

=

|

α

|

±

|

β

|

.
Also, if

β

k

≤

α

k

for all

k

=

1

,

…

,

n

, then we write

β

≤

α

. For multi-indices

α

,

β

, with

β

≤

α

, we define

(

α

β

)

=

α

!

(

α

-

β

)

!

⁢

β

!

.

For a point

x

=

(

x

1

,

…

,

x

n

)

in

ℝ

n

(with
standard coordinates) we define

x

α

=

∏

k

=

1

n

x

k

α

k

.

Also, if

f

:

ℝ

n

→

ℝ

is a

smooth function

, and

α

=

(

α

1

,

…

,

α

n

)

is a multi-index, we define

∂

α

⁡

f

=

∂

|

α

|

∂

α

1

⁡

e

1

⁢

⋯

⁢

∂

α

n

⁡

e

n

⁢

f

,

where

e

1

,

…

,

e

n

are the standard unit vectors of

ℝ

n

.
Since

f

is sufficiently smooth, the order in which the

derivations

are
performed is irrelevant. For multi-indices

α

and

β

, we thus
have

∂

α

⁡

∂

β

=

∂

α

+

β

=

∂

β

+

α

=

∂

β

⁡

∂

α

.

Examples

1.

If

n

is a

positive integer

, and

x

1

,

…

,

x

k

are

complex numbers

, the multinomial

expansion

states that

(

x

1

+

⋯

+

x

k

)

n

=

n

!

⁢

∑

|

α

|

=

n

x

α

α

!

,

where

x

=

(

x

1

,

…

,

x

k

)

and

α

is a multi-index.
(proof (

http://planetmath.org/MultinomialTheoremProof

))

2.

Leibniz’ rule: If

f

,

g

:

ℝ

n

→

ℝ

are smooth functions, and

β

is
a multi-index, then

∂

β

⁡

(

f

⁢

g

)

=

∑

α

≤

β

(

β

α

)

⁢

∂

α

⁡

(

f

)

⁢

∂

β

-

α

⁡

(

g

)

,

where

α

is a multi-index.

References

1

M. Reed, B. Simon,

Methods of Mathematical Physics,
I -

Functional Analysis

, Academic Press, 1980.

Title

multi-index notation

Canonical name

MultiindexNotation

Date of creation

2013-03-22 13:41:32

Last modified on

2013-03-22 13:41:32

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

15

Author

matte (1858)

Entry type

Definition

Classification

msc 05-00

Defines

multi-index

Defines

multi-indices

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