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Difference between revisions of "Multi-index notation"

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$\def\a{\alpha}$
 
$\def\a{\alpha}$
 
$\def\b{\beta}$
 
$\def\b{\beta}$
  
 
An abbreviated form of notation in analysis, imitating the vector notation by single letters rather than by listing all vector components.
 
An abbreviated form of notation in analysis, imitating the vector notation by single letters rather than by listing all vector components.
== Monomials ==
+
----
A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field) is denoted by $x$. For a ''multiindex'' $\a=(\a_1,\dots,\a_n)\in\Z_+^n$ the expression $x^\a$ denotes the product, $x_\a=x_1^{\a_1}\cdots x_n^{\a_n}$. Other expressions related to multiindices are expanded as follows:
+
==Rules==
 +
A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field $\Bbbk$) is denoted by $x$. For a ''multiindex'' $\a=(\a_1,\dots,\a_n)\in\Z_+^n$ the expression $x^\a$ denotes the product, $x_\a=x_1^{\a_1}\cdots x_n^{\a_n}$. Other expressions related to multiindices are expanded as follows:
 
$$
 
$$
 
\begin{aligned}
 
\begin{aligned}
 
|\a|&=\a_1+\cdots+\a_n\in\Z_+^n,
 
|\a|&=\a_1+\cdots+\a_n\in\Z_+^n,
 
\\
 
\\
\a!&=\a_1!\cdots\a_n!\qquad\text{(as usual,}0!=1!=1),
+
\a!&=\a_1!\cdots\a_n!\qquad\text{(as usual, }0!=1!=1),
 
\\
 
\\
x^\a&=x_1^{\a_1}\cdots x_n^{\a_n},
+
x^\a&=x_1^{\a_1}\cdots x_n^{\a_n}\in \Bbbk[x]=\Bbbk[x_1,\dots,x_n],
 
\\
 
\\
 
\a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n,
 
\a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n,
 
\end{aligned}
 
\end{aligned}
 
$$
 
$$
 +
The convention extends for the binomial coefficients ($\a\geqslant\b$ means, quite naturally, that $\a_1\geqslant\b_1,\dots,\a_n\geqslant\b_n$):
 +
$$
 +
\binom{\a}{\b}=\binom{\a_1}{\b_1}\cdots\binom{\a_n}{\b_n}=\frac{\a!}{\b!(\a-\b)!},\qquad \text{if}\quad \a\geqslant\b.
 +
$$
 +
The partial derivative operators are also abbreviated:
 +
$$
 +
\partial_x=\biggl(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}\biggr)=\partial\quad\text{if the choice of $x$ is clear from context.}
 +
$$
 +
The notation for partial derivatives is also quite natural: for a differentiable function $f(x_1,\dots,x_n)$ of $n$ variables,
 +
$$
 +
\partial^a f=\frac{\partial^{|\a|} f}{\partial x^\a}=\frac{\partial^{\a_1}}{\partial x_1^{\a_1}}\cdots\frac{\partial^{\a_n}}{\partial x_n^{\a_n}}f=\frac{\partial^{|\a|}f}{\partial x_1^{\a_1}\cdots\partial x_n^{\a_n}}.
 +
$$
 +
If $f$ is itself a vector-valued function of dimension $m$, the above partial derivatives are $m$-vectors. The notation
 +
$$
 +
\partial f=\bigg(\frac{\partial f}{\partial x}\bigg)
 +
$$
 +
is used to denote the Jacobian matrix of a function $f$ (in general, only rectangular).
 +
 +
;Caveat
 +
The notation $\a>0$ is ambiguous, especially in mathematical economics, as it may either mean that $\a_1>0,\dots,\a_n>0$, or $0\ne\a\geqslant0$.
 +
 +
==Examples==
 +
===Binomial formula===
 +
$$
 +
(x+y)^\a=\sum_{0\leqslant\b\leqslant\a}\binom\a\b x^{\a-\b} y^\b.
 +
$$
 +
===Leibniz formula for higher derivatives of multivariate functions===
 +
$$
 +
\partial^\a(fg)=\sum_{0\leqslant\b\leqslant\a}\binom\a\b \partial^{\a-\b}f\cdot \partial^\b g.
 +
$$
 +
In particular,
 +
$$
 +
\partial^\a x^\beta=\begin{cases}
 +
\frac{\b!}{(\b-\a)!}x^{\b-\a},\qquad&\text{if }\a\leqslant\b,
 +
\\
 +
\quad 0,&\text{otherwise}.
 +
\end{cases}
 +
$$
 +
 +
===Taylor series of a smooth function===
 +
If $f$ is infinitely smooth near the origin $x=0$, then its Taylor series (at the origin) has the form
 +
$$
 +
\sum_{\a\in\Z_+^n}\frac1{\a!}\partial^\a f(0)\cdot x^\a.
 +
$$
 +
===Symbol of a differential operator===
 +
If
 +
$$
 +
D=\sum_{|\a|\le d}a_\a(x)\partial^\a
 +
$$
 +
is a linear ordinary differential operator with variable coefficients $a_\a(x)$, then its ''principal symbol'' is the function of $2n$ variables $S(x,p)=\sum_{|\a|=d}a_\a(x)p^\a$.

Latest revision as of 11:12, 12 December 2013


$\def\a{\alpha}$ $\def\b{\beta}$

An abbreviated form of notation in analysis, imitating the vector notation by single letters rather than by listing all vector components.


Rules

A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field $\Bbbk$) is denoted by $x$. For a multiindex $\a=(\a_1,\dots,\a_n)\in\Z_+^n$ the expression $x^\a$ denotes the product, $x_\a=x_1^{\a_1}\cdots x_n^{\a_n}$. Other expressions related to multiindices are expanded as follows: $$ \begin{aligned} |\a|&=\a_1+\cdots+\a_n\in\Z_+^n, \\ \a!&=\a_1!\cdots\a_n!\qquad\text{(as usual, }0!=1!=1), \\ x^\a&=x_1^{\a_1}\cdots x_n^{\a_n}\in \Bbbk[x]=\Bbbk[x_1,\dots,x_n], \\ \a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n, \end{aligned} $$ The convention extends for the binomial coefficients ($\a\geqslant\b$ means, quite naturally, that $\a_1\geqslant\b_1,\dots,\a_n\geqslant\b_n$): $$ \binom{\a}{\b}=\binom{\a_1}{\b_1}\cdots\binom{\a_n}{\b_n}=\frac{\a!}{\b!(\a-\b)!},\qquad \text{if}\quad \a\geqslant\b. $$ The partial derivative operators are also abbreviated: $$ \partial_x=\biggl(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}\biggr)=\partial\quad\text{if the choice of $x$ is clear from context.} $$ The notation for partial derivatives is also quite natural: for a differentiable function $f(x_1,\dots,x_n)$ of $n$ variables, $$ \partial^a f=\frac{\partial^{|\a|} f}{\partial x^\a}=\frac{\partial^{\a_1}}{\partial x_1^{\a_1}}\cdots\frac{\partial^{\a_n}}{\partial x_n^{\a_n}}f=\frac{\partial^{|\a|}f}{\partial x_1^{\a_1}\cdots\partial x_n^{\a_n}}. $$ If $f$ is itself a vector-valued function of dimension $m$, the above partial derivatives are $m$-vectors. The notation $$ \partial f=\bigg(\frac{\partial f}{\partial x}\bigg) $$ is used to denote the Jacobian matrix of a function $f$ (in general, only rectangular).

Caveat

The notation $\a>0$ is ambiguous, especially in mathematical economics, as it may either mean that $\a_1>0,\dots,\a_n>0$, or $0\ne\a\geqslant0$.

Examples

Binomial formula

$$ (x+y)^\a=\sum_{0\leqslant\b\leqslant\a}\binom\a\b x^{\a-\b} y^\b. $$

Leibniz formula for higher derivatives of multivariate functions

$$ \partial^\a(fg)=\sum_{0\leqslant\b\leqslant\a}\binom\a\b \partial^{\a-\b}f\cdot \partial^\b g. $$ In particular, $$ \partial^\a x^\beta=\begin{cases} \frac{\b!}{(\b-\a)!}x^{\b-\a},\qquad&\text{if }\a\leqslant\b, \\ \quad 0,&\text{otherwise}. \end{cases} $$

Taylor series of a smooth function

If $f$ is infinitely smooth near the origin $x=0$, then its Taylor series (at the origin) has the form $$ \sum_{\a\in\Z_+^n}\frac1{\a!}\partial^\a f(0)\cdot x^\a. $$

Symbol of a differential operator

If $$ D=\sum_{|\a|\le d}a_\a(x)\partial^\a $$ is a linear ordinary differential operator with variable coefficients $a_\a(x)$, then its principal symbol is the function of $2n$ variables $S(x,p)=\sum_{|\a|=d}a_\a(x)p^\a$.

How to Cite This Entry:
Multi-index notation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Multi-index_notation&oldid=25750