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 non-negative integer index (in the sense of exponent) 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 $\mathbb {N} _{0}^{n}$). 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 $|\alpha |=k\,$

• 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 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

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

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

multi-index derivative of a power at PlanetMath.