Multi-index notation

The mathematical notation of multi-indices simplifies formulae used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to a vector of indices.

Multi-index notation

An n-dimensional multi-index is a vector

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

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

Componentwise sum and difference

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

Partial order

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

{\alpha  \choose \beta }={\alpha _{1} \choose \beta _{1}}{\alpha _{2} \choose \beta _{2}}\cdots {\alpha _{n} \choose \beta _{n}}

Power

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

Higher-order partial derivative

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

where

\partial _{i}^{\alpha _{i}}:=\partial ^{\alpha _{i}}/\partial x_{i}^{\alpha _{i}}

Some applications

The multi-index notation allows to extend many formulae from elementary calculus to the corresponding multi-variable case. Here are some examples:

Multinomial theorem

{\biggl (}\sum _{i=1}^{n}x_{i}{\biggr )}^{k}=\sum _{|\alpha |=k}{\frac {k!}{\alpha !}}\,x^{\alpha }

Leibniz formula

For smooth functions f and g

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

Taylor series

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 |\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

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 partial differential operator

A formal N-th order partial differential operator in n variables is written as

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

Integration by parts

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

\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.

An example theorem

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

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

Proof

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

{\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}}\qquad (1)

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

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