Multi-index notation

An n-dimensional multi-index is an ${\textstyle n}$-tuple

${\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}$

of non-negative integers (i.e. an element of the ${\textstyle n}$-dimensional set of natural numbers, denoted ${\displaystyle \mathbb {N} _{0}^{n}}$).

For multi-indices ${\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}}$ and ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}}$, one defines:

Componentwise sum and difference

${\displaystyle \alpha \pm \beta =(\alpha _{1}\pm \beta _{1},\,\alpha _{2}\pm \beta _{2},\ldots ,\,\alpha _{n}\pm \beta _{n})}$

Partial order

${\displaystyle \alpha \leq \beta \quad \Leftrightarrow \quad \alpha _{i}\leq \beta _{i}\quad \forall \,i\in \{1,\ldots ,n\}}$

Sum of components (absolute value)

${\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}$

Factorial

${\displaystyle \alpha$ !=\alpha _{1}!\cdot \alpha _{2}!\cdots \alpha _{n}!}

Binomial coefficient

${\displaystyle {\binom {\alpha }{\beta }}={\binom {\alpha _{1}}{\beta _{1}}}{\binom {\alpha _{2}}{\beta _{2}}}\cdots {\binom {\alpha _{n}}{\beta _{n}}}={\frac {\alpha$ !}{\beta !(\alpha -\beta )!}}}

Multinomial coefficient

$${\displaystyle {\binom {k}{\alpha }}={\frac {k!}{\alpha _{1}!\alpha _{2}!\cdots \alpha _{n}!}}={\frac {k!}{\alpha$$ !}}}

where ${\displaystyle k:=|\alpha |\in \mathbb {N} _{0}}$.

Power

${\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}}$.

Higher-order partial derivative

$${\displaystyle \partial ^{\alpha }=\partial _{1}^{\alpha _{1}}\partial _{2}^{\alpha _{2}}\ldots \partial _{n}^{\alpha _{n}},}$$

where ${\displaystyle \partial _{i}^{\alpha _{i}}:=\partial ^{\alpha _{i}}/\partial x_{i}^{\alpha _{i}}}$ (see also 4-gradient). Sometimes the notation ${\displaystyle D^{\alpha }=\partial ^{\alpha }}$ is also used.[1]

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, ${\displaystyle x,y,h\in \mathbb {C} ^{n}}$ (or ${\displaystyle \mathbb {R} ^{n}}$), ${\displaystyle \alpha ,\nu \in \mathbb {N} _{0}^{n}}$, and ${\displaystyle f,g,a_{\alpha }\colon \mathbb {C} ^{n}\to \mathbb {C} }$ (or ${\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }$).

Multinomial theorem

${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)^{k}=\sum _{|\alpha |=k}{\binom {k}{\alpha }}\,x^{\alpha }}$

Multi-binomial theorem

$${\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,x^{\nu }y^{\alpha -\nu }.}$$

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⋯(x_n + y_n)^α_n.

Leibniz formula

For smooth functions ${\textstyle f}$ and ${\textstyle g}$,

$${\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,\partial ^{\nu }f\,\partial ^{\alpha -\nu }g.}$$

Taylor series

For an analytic function ${\textstyle f}$ in ${\textstyle n}$ variables one has

$${\displaystyle f(x+h)=\sum _{\alpha \in \mathbb {N} _{0}^{n}}{{\frac {\partial ^{\alpha }f(x)}{\alpha$$ !}}h^{\alpha }}.}

In fact, for a smooth enough function, we have the similar Taylor expansion

$${\displaystyle f(x+h)=\sum _{|\alpha |\leq n}{{\frac {\partial ^{\alpha }f(x)}{\alpha$$ !}}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

$${\displaystyle R_{n}(x,h)=(n+1)\sum _{|\alpha |=n+1}{\frac {h^{\alpha }}{\alpha$$ !}}\int _{0}^{1}(1-t)^{n}\partial ^{\alpha }f(x+th)\,dt.}

General linear partial differential operator

A formal linear ${\textstyle N}$-th order partial differential operator in ${\textstyle n}$ variables is written as

$${\displaystyle P(\partial )=\sum _{|\alpha |\leq N}{a_{\alpha }(x)\partial ^{\alpha }}.}$$

Integration by parts

For smooth functions with compact support in a bounded domain ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ one has

$${\displaystyle \int _{\Omega }u(\partial ^{\alpha }v)\,dx=(-1)^{|\alpha |}\int _{\Omega }{(\partial ^{\alpha }u)v\,dx}.}$$

This formula is used for the definition of distributions and weak derivatives.

If ${\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}}$ are multi-indices and ${\displaystyle x=(x_{1},\ldots ,x_{n})}$, then

$${\displaystyle \partial ^{\alpha }x^{\beta }={\begin{cases}{\frac {\beta$$ !}{(\beta -\alpha )!}}x^{\beta -\alpha }&{\text{if}}~\alpha \leq \beta ,\\0&{\text{otherwise.}}\end{cases}}}

Proof

The proof follows from the power rule for the ordinary derivative; if α and β are in ${\textstyle \{0,1,2,\ldots \}}$, then

$${\displaystyle {\frac {d^{\alpha }}{dx^{\alpha }}}x^{\beta }={\begin{cases}{\frac {\beta$$ !}{(\beta -\alpha )!}}x^{\beta -\alpha }&{\hbox{if}}\,\,\alpha \leq \beta ,\\0&{\hbox{otherwise.}}\end{cases}}}

(1)

Suppose ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}$, ${\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})}$, and ${\displaystyle x=(x_{1},\ldots ,x_{n})}$. Then we have that

$${\displaystyle {\begin{aligned}\partial ^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}}$$

For each ${\textstyle i}$ in ${\textstyle \{1,\ldots ,n\}}$, the function ${\displaystyle x_{i}^{\beta _{i}}}$ only depends on ${\displaystyle x_{i}}$. In the above, each partial differentiation ${\displaystyle \partial /\partial x_{i}}$ therefore reduces to the corresponding ordinary differentiation ${\displaystyle d/dx_{i}}$. Hence, from equation (1), it follows that ${\displaystyle \partial ^{\alpha }x^{\beta }}$ vanishes if ${\textstyle \alpha _{i}>\beta _{i}}$ for at least one ${\textstyle i}$ in ${\textstyle \{1,\ldots ,n\}}$. If this is not the case, i.e., if ${\textstyle \alpha \leq \beta }$ as multi-indices, then

$${\displaystyle {\frac {d^{\alpha _{i}}}{dx_{i}^{\alpha _{i}}}}x_{i}^{\beta _{i}}={\frac {\beta _{i}!}{(\beta _{i}-\alpha _{i})!}}x_{i}^{\beta _{i}-\alpha _{i}}}$$

for each ${\displaystyle i}$ and the theorem follows. Q.E.D.

• Einstein notation
• Index notation
• Ricci calculus

• 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.