Multi-index notation

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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 an ordered tuple of indices.

Multi-index notation

An n-dimensional multi-index is an n-tuple

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

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 $x=(x_{1},x_{2},\ldots ,x_{n})\in {\mathbb {R}}^{n}$ 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: ${\binom {\alpha }{\beta }}={\binom {\alpha _{1}}{\beta _{1}}}{\binom {\alpha _{2}}{\beta _{2}}}\cdots {\binom {\alpha _{n}}{\beta _{n}}}={\frac {\alpha !}{\beta !(\alpha -\beta )!}}$

Multinomial coefficient: ${\binom {k}{\alpha }}={\frac {k!}{\alpha _{1}!\alpha _{2}!\cdots \alpha _{n}!}}={\frac {k!}{\alpha !}}$

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

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

Higher-order partial derivative

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

where $\partial _{i}^{{\alpha _{i}}}:=\partial ^{{\alpha _{i}}}/\partial x_{i}^{{\alpha _{i}}}$ (see also 4-gradient).

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

Multinomial theorem

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

Multi-binomial theorem

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

Note that, since $x + y$ is a vector and $α$ is a multi-index, the expression on the left is short for $(x 1 + y 1) α 1 ...(x n + y n) α n$ .

Leibniz formula

For smooth functions f and g

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

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

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

for each $i$ and the theorem follows. $\Box$

References

Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. ISBN 0-8493-7158-9

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Multi-index notation