Multi-index notation

Template:Short description Multi-index notation is a mathematical notation that simplifies formulas 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

$\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$

of non-negative integers (i.e. an element of the $n$-dimensional set of natural numbers, denoted ${\mathbb{N}}_0^n$).

For multi-indices $\alpha ,\beta \in {\mathbb{N}}_0^n$ and $x=(x_1,x_2,\dots ,x_n)\in {ℝ}^n$, one defines:

Componentwise sum and difference

$\alpha \pm \beta =({\alpha }_1\pm {\beta }_1,{\alpha }_2\pm {\beta }_2,\dots ,{\alpha }_n\pm {\beta }_n)$

Partial order

$\alpha \le \beta \iff {\alpha }_i\le {\beta }_i\mathrm{\forall }i\in \{1,\dots ,n\}$

Sum of components (absolute value)

$|\alpha |={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n$

Factorial

$\alpha !={\alpha }_1!\cdot {\alpha }_2!\cdots {\alpha }_n!$

Binomial coefficient

$(\genfrac{}{}{0ex}{}{\alpha }{\beta })=(\genfrac{}{}{0ex}{}{{\alpha }_1}{{\beta }_1})(\genfrac{}{}{0ex}{}{{\alpha }_2}{{\beta }_2})\cdots (\genfrac{}{}{0ex}{}{{\alpha }_n}{{\beta }_n})=\frac{\alpha !}{\beta !(\alpha -\beta )!}$

Multinomial coefficient

$$(\genfrac{}{}{0ex}{}{k}{\alpha })=\frac{k!}{{\alpha }_1!{\alpha }_2!\cdots {\alpha }_n!}=\frac{k!}{\alpha !}$$ where $k:=|\alpha |\in {\mathbb{N}}_0$.

Power

$x^{\alpha }=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\dots x_n^{{\alpha }_n}$.

Higher-order partial derivative

$${\partial }^{\alpha }={\partial }_1^{{\alpha }_1}{\partial }_2^{{\alpha }_2}\dots {\partial }_n^{{\alpha }_n},$$ where ${\partial }_i^{{\alpha }_i}:={\partial }^{{\alpha }_i}/\partial x_i^{{\alpha }_i}$ (see also 4-gradient). Sometimes the notation $D^{\alpha }={\partial }^{\alpha }$ is also used.

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, $x,y,h\in {ℂ}^n$ (or ${ℝ}^n$), and $\alpha ,\beta ,\nu \in {\mathbb{N}}_0^n$.

Derivative of a monomial

$${\partial }^{\alpha }{x}^{\beta }=\{\begin{array}{cc}\frac{\beta !}{(\beta -\alpha )!}{x}^{\beta -\alpha },& \text{if}\alpha \le \beta ,\\ 0,& \text{otherwise.}\end{array}$$

Multinomial theorem

${\left(\sum _{i=1}^n{x}_i\right)}^k=\sum _{|\alpha |=k}(\genfrac{}{}{0ex}{}{k}{\alpha })x^{\alpha }.$

Multi-binomial theorem

Template:Main $$(x+y{)}^{\alpha }=\sum _{\nu \le \alpha }(\genfrac{}{}{0ex}{}{\alpha }{\nu })x^{\nu }{y}^{\alpha -\nu }.$$ Note that, since Template:Math is a vector and Template:Math is a multi-index, the expression on the left is short for Template:Math.

Leibniz formula

Template:Main For smooth functions $f$ and $g$,$${\partial }^{\alpha }(fg)=\sum _{\nu \le \alpha }(\genfrac{}{}{0ex}{}{\alpha }{\nu }){\partial }^{\nu }f{\partial }^{\alpha -\nu }g.$$

Taylor series

Template:Main For an analytic function $f$ in $n$ variables one has $$f(x+h)=\sum _{\alpha \in {\mathbb{N}}_0^n}\frac{{\partial }^{\alpha }f(x)}{\alpha !}{h}^{\alpha }.$$ In fact, for a smooth enough function, we have the similar Taylor expansion $$f(x+h)=\sum _{|\alpha |\le n}\frac{{\partial }^{\alpha }f(x)}{\alpha !}{h}^{\alpha }+R_n(x,h),$$ 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 $$R_n(x,h)=(n+1)\sum _{|\alpha |=n+1}\frac{h^{\alpha }}{\alpha !}{\int }_0^1(1-t{)}^n{\partial }^{\alpha }f(x+th)dt.$$

Integration by parts

Template:Main For smooth functions $u$ and $v$ with compact support in ${ℝ}^n$, $${\int }_{{ℝ}^n}u({\partial }^{\alpha }v)dx=(-1{)}^{|\alpha |}{\int }_{{ℝ}^n}({\partial }^{\alpha }u)vdx.$$ This formula is used for the definition of weak derivatives of distributions.

See also

• Einstein notation
• Index notation
• Ricci calculus

References

• Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. Template:Isbn