Multi-index notation

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

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

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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 $S(x,p)=\sum_{|\a|=d}a_\a(x)p^\a$.