# Difference between revisions of "Multi-index notation"

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$\def\b{\beta}$ | $\def\b{\beta}$ | ||

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. | ||

− | == | + | ---- |

− | 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 09:12, 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.

## Contents

## 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://encyclopediaofmath.org/index.php?title=Multi-index_notation&oldid=25750