Difference between revisions of "Multi-index notation"

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en>Magioladitis
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{{Calculus|expanded=Multivariable calculus}}
 
{{Calculus|expanded=Multivariable calculus}}
The [[mathematical notation]] of '''multi-indices''' simplifies formulae used in [[multivariable calculus]], [[partial differential equation]]s and the theory of [[distribution (mathematics)|distribution]]s, by generalising the concept of an integer [[index notation|index]] to an ordered [[tuple]] of indices.  
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'''Multi-index notation''' is a [[mathematical notation]] that simplifies formulae used in [[multivariable calculus]], [[partial differential equation]]s and the theory of [[distribution (mathematics)|distribution]]s, by generalising the concept of an integer [[index notation|index]] to an ordered [[tuple]] of indices.
 +
 
 +
==Definition and basic properties==
  
==Multi-index notation==
 
 
An ''n''-dimensional '''multi-index''' is an ''n''-[[tuple]]
 
An ''n''-dimensional '''multi-index''' is an ''n''-[[tuple]]
  
 
:<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>
 
:<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>
  
of [[natural number|non-negative integers]] (i.e. an element of the ''n''-[[dimension]]al [[set (mathematics)|set]] of [[natural number]]s, denoted <math>\mathbb{N}^n_0</math>).  
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of [[natural number|non-negative integers]] (i.e. an element of the ''n''-[[dimension]]al [[set (mathematics)|set]] of [[natural number]]s, denoted <math>\mathbb{N}^n_0</math>).
  
 
For multi-indices <math>\alpha, \beta \in \mathbb{N}^n_0</math> and <math>x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n</math> one defines:
 
For multi-indices <math>\alpha, \beta \in \mathbb{N}^n_0</math> and <math>x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n</math> one defines:
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;[[Multinomial coefficient]]
 
;[[Multinomial coefficient]]
:<math>\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} </math>  
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:<math>\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} </math>
  
 
where <math>k:=|\alpha|\in\mathbb{N}_0\,\!</math>.
 
where <math>k:=|\alpha|\in\mathbb{N}_0\,\!</math>.
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;Higher-order [[partial derivative]]
 
;Higher-order [[partial derivative]]
  
:<math>\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n}</math>  
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:<math>\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n}</math>
 
where <math>\partial_i^{\alpha_i}:=\part^{\alpha_i} / \part x_i^{\alpha_i}</math> (see also [[4-gradient]]).
 
where <math>\partial_i^{\alpha_i}:=\part^{\alpha_i} / \part x_i^{\alpha_i}</math> (see also [[4-gradient]]).
  
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;[[Integration by parts]]
 
;[[Integration by parts]]
For smooth functions with [[compact support]] in a bounded domain <math>\Omega \subset \mathbb{R}^n</math> one has  
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For smooth functions with [[compact support]] in a bounded domain <math>\Omega \subset \mathbb{R}^n</math> one has
  
 
:<math>\int_{\Omega}{}{u(\partial^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(\partial^{\alpha}u)v\,dx}.</math>
 
:<math>\int_{\Omega}{}{u(\partial^{\alpha}v)}\,dx = (-1)^{|\alpha|}\int_{\Omega}^{}{(\partial^{\alpha}u)v\,dx}.</math>
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:<math> \part^\alpha x^\beta =  
 
:<math> \part^\alpha x^\beta =  
 
\begin{cases}  
 
\begin{cases}  
\frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \hbox{if}\,\, \alpha\le\beta,\\  
+
\frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \hbox{if}\,\, \alpha\le\beta,\\
 
  0 & \hbox{otherwise.} \end{cases}</math>
 
  0 & \hbox{otherwise.} \end{cases}</math>
  
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:<math>\begin{align}\part^\alpha x^\beta&= \frac{\part^{\vert\alpha\vert}}{\part x_1^{\alpha_1} \cdots \part x_n^{\alpha_n}} x_1^{\beta_1} \cdots x_n^{\beta_n}\\
 
:<math>\begin{align}\part^\alpha x^\beta&= \frac{\part^{\vert\alpha\vert}}{\part x_1^{\alpha_1} \cdots \part x_n^{\alpha_n}} x_1^{\beta_1} \cdots x_n^{\beta_n}\\
 
&= \frac{\part^{\alpha_1}}{\part x_1^{\alpha_1}} x_1^{\beta_1} \cdots
 
&= \frac{\part^{\alpha_1}}{\part x_1^{\alpha_1}} x_1^{\beta_1} \cdots
\frac{\part^{\alpha_n}}{\part x_n^{\alpha_n}} x_n^{\beta_n}.\end{align}</math>
+
\frac{\part^{\alpha_n}}{\part x_n^{\alpha_n}} x_n^{\beta_n}.\end{align}</math>
  
 
For each ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}, the function <math>x_i^{\beta_i}</math> only depends on <math>x_i</math>. In the above, each partial differentiation <math>\part/\part x_i</math> therefore reduces to the corresponding ordinary differentiation <math>d/dx_i</math>. Hence, from equation (1), it follows that <math>\part^\alpha x^\beta</math> vanishes if ''&alpha;<sub>i</sub>''&nbsp;>&nbsp;''&beta;<sub>i</sub>'' for at least one ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}. If this is not the case, i.e., if ''&alpha;''&nbsp;&le;&nbsp;''&beta;'' as multi-indices, then
 
For each ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}, the function <math>x_i^{\beta_i}</math> only depends on <math>x_i</math>. In the above, each partial differentiation <math>\part/\part x_i</math> therefore reduces to the corresponding ordinary differentiation <math>d/dx_i</math>. Hence, from equation (1), it follows that <math>\part^\alpha x^\beta</math> vanishes if ''&alpha;<sub>i</sub>''&nbsp;>&nbsp;''&beta;<sub>i</sub>'' for at least one ''i'' in {1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''n''}. If this is not the case, i.e., if ''&alpha;''&nbsp;&le;&nbsp;''&beta;'' as multi-indices, then

Latest revision as of 06:53, 14 August 2014

{{#invoke: Sidebar | collapsible }} Multi-index notation is a mathematical notation that simplifies formulae 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 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 applications

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 partial differential operator

A formal 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

If are multi-indices 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 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 also

References

  • 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.