Multi-index notation

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

An n-dimensional multi-index is a vector

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

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

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

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

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

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

{\alpha  \choose \beta }={\frac {\alpha !}{(\alpha -\beta )!\,\beta !}}={\alpha _{1} \choose \beta _{1}}{\alpha _{2} \choose \beta _{2}}\cdots {\alpha _{n} \choose \beta _{n}}

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

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

where

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

The notation allows to extend many formula from elementary calculus to the corresponding multi-variable case. Some examples of common applications of multi-index notations:

Multinomial expansion:

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

Leibniz formula: for smooth functions u, v

\partial ^{\alpha }(uv)=\sum _{\nu \leq \alpha }^{}{{\alpha  \choose \nu }\partial ^{\nu }u\,\partial ^{\alpha -\nu }v}

Taylor series: for an analytic function f one has

f(\mathbf {x} +\mathbf {h} )=\sum _{|\alpha |\geq 0}^{}{{\frac {\partial ^{\alpha }f(\mathbf {x} )}{\alpha !}}\mathbf {h} ^{\alpha }}

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

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

where the last term (the remainder) depends on the exact version of the Taylor formula. for instance, for the Cauchy formula (with integral remainder), one gets

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

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 }}

Partial integration: 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.

Theorem

Theorem If $i,k$ are multi-indices in $\mathbb {N} ^{n}$ , and $x=(x_{1},\ldots ,x_{n})$ , then

\partial ^{i}x^{k}=\left\{{\begin{matrix}{\frac {k!}{(k-i)!}}x^{k-i}&{\hbox{if}}\,\,i\leq k\\0&{\hbox{otherwise.}}\end{matrix}}\right.

Proof. The proof follows from the corresponding rule for the ordinary derivative; if $i,k$ are in $0,1,2,\ldots$ , then

{\frac {d^{i}}{dx^{i}}}x^{k}=\left\{{\begin{matrix}{\frac {k!}{(k-i)!}}x^{k-i}&{\hbox{if}}\,\,i\leq k,\\0&{\hbox{otherwise.}}\end{matrix}}\right.

. (1)

Suppose $i=(i_{1},\ldots ,i_{n})$ , $k=(k_{1},\ldots ,k_{n})$ , and $x=(x_{1},\ldots ,x_{n})$ . Then we have that

\partial ^{i}x^{k}

=

{\frac {\partial ^{\vert i\vert }}{\partial x_{1}^{i_{1}}\cdots \partial x_{n}^{i_{n}}}}x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}

=

{\frac {\partial ^{i_{1}}}{\partial x_{1}^{i_{1}}}}x_{1}^{k_{1}}\cdots {\frac {\partial ^{i_{n}}}{\partial x_{n}^{i_{n}}}}x_{n}^{k_{n}}

.

For each $r=1,\ldots ,n$ , the function $x_{r}^{k_{r}}$ only depends on $x_{r}$ . In the above, each partial differentiation $\partial /\partial x_{r}$ therefore reduces to the corresponding ordinary differentiation $d/dx_{r}$ . Hence, from equation 1, it follows that $\partial ^{i}x^{k}$ vanishes if $i_{r}>k_{r}$ for any $r=1,\ldots ,n$ . If this is not the case, i.e., if $i\leq k$ as multi-indices, then for each $r$ ,