Multi-index notation

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}} Multi-index notation is a mathematical notation that simplifies formulas 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.

Definition and basic propertiesEdit

An n-dimensional multi-index is an <math display="inline">n</math>-tuple

<math>\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n)</math>

of non-negative integers (i.e. an element of the <math display="inline">n</math>-dimensional set of natural numbers, denoted <math>\mathbb{N}^n_0</math>).

For multi-indices <math>\alpha, \beta \in \mathbb{N}^n_0</math> and <math>x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n</math>, one defines:

Componentwise sum and difference: <math>\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n)</math>
Partial order: <math>\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\{1,\ldots,n\}</math>
Sum of components (absolute value): <math>| \alpha | = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>
Factorial: <math>\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n!</math>
Binomial coefficient: <math>\binom{\alpha}{\beta} = \binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots\binom{\alpha_n}{\beta_n} = \frac{\alpha!}{\beta!(\alpha-\beta)!}</math>
Multinomial coefficient: <math display="block">\binom{k}{\alpha} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_n! } = \frac{k!}{\alpha!} </math> where <math>k:=|\alpha|\in\mathbb{N}_0</math>.
Power: <math>x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \ldots x_n^{\alpha_n}</math>.
Higher-order partial derivative: <math display="block">\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_n^{\alpha_n},</math> where <math>\partial_i^{\alpha_i}:=\partial^{\alpha_i} / \partial x_i^{\alpha_i}</math> (see also 4-gradient). Sometimes the notation <math>D^{\alpha} = \partial^{\alpha}</math> is also used.<ref>Template:Cite book</ref>

Some applicationsEdit

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, <math>x,y,h\in\Complex^n</math> (or <math>\R^n</math>), <math>\alpha,\nu\in\N_0^n</math>, and <math>f,g,a_\alpha\colon\Complex^n\to\Complex</math> (or <math>\R^n\to\R</math>).

Multinomial theorem: <math> \left( \sum_{i=1}^n x_i\right)^k = \sum_{|\alpha|=k} \binom{k}{\alpha} \, x^\alpha</math>
Multi-binomial theorem: <math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, x^\nu y^{\alpha - \nu}.</math> Note that, since Template:Math is a vector and Template:Math is a multi-index, the expression on the left is short for Template:Math.
Leibniz formula: For smooth functions <math display="inline">f</math> and <math display="inline">g</math>,<math display="block">\partial^\alpha(fg) = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} \, \partial^{\nu}f\,\partial^{\alpha-\nu}g.</math>
Taylor series: For an analytic function <math display="inline">f</math> in <math display="inline">n</math> variables one has <math display="block">f(x+h) = \sum_{\alpha\in\mathbb{N}^n_0} {\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}.</math> In fact, for a smooth enough function, we have the similar Taylor expansion <math display="block">f(x+h) = \sum_{|\alpha| \le n}{\frac{\partial^{\alpha}f(x)}{\alpha !}h^\alpha}+R_{n}(x,h),</math> 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 <math display="block">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.</math>
General linear partial differential operator: A formal linear <math display="inline">N</math>-th order partial differential operator in <math display="inline">n</math> variables is written as <math display="block">P(\partial) = \sum_{|\alpha| \le N} {a_{\alpha}(x)\partial^{\alpha}}.</math>
Integration by parts: For smooth functions with compact support in a bounded domain <math>\Omega \subset \R^n</math> one has <math display="block">\int_{\Omega} u(\partial^{\alpha}v) \, dx = (-1)^{|\alpha|} \int_{\Omega} {(\partial^{\alpha}u)v\,dx}.</math> This formula is used for the definition of distributions and weak derivatives.

An example theoremEdit

If <math>\alpha,\beta\in\mathbb{N}^n_0</math> are multi-indices and <math>x=(x_1,\ldots, x_n)</math>, then <math display="block"> \partial^\alpha x^\beta = \begin{cases} \frac{\beta!}{(\beta-\alpha)!} x^{\beta-\alpha} & \text{if}~ \alpha\le\beta,\\ 0 & \text{otherwise.} \end{cases}</math>

ProofEdit

The proof follows from the power rule for the ordinary derivative; if α and β are in <math display="inline">\{0, 1, 2,\ldots\}</math>, then Template:NumBlk

Suppose <math>\alpha=(\alpha_1,\ldots, \alpha_n)</math>, <math>\beta=(\beta_1,\ldots, \beta_n)</math>, and <math>x=(x_1,\ldots, x_n)</math>. Then we have that <math display="block">\begin{align}\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{align}</math>

For each <math display="inline">i</math> in <math display="inline">\{ 1, \ldots , n\}</math>, the function <math>x_i^{\beta_i}</math> only depends on <math>x_i</math>. In the above, each partial differentiation <math>\partial/\partial x_i</math> therefore reduces to the corresponding ordinary differentiation <math>d/dx_i</math>. Hence, from equation (Template:EquationNote), it follows that <math>\partial^\alpha x^\beta</math> vanishes if <math display="inline">\alpha_i > \beta_i</math> for at least one <math display="inline">i</math> in <math display="inline">\{ 1, \ldots , n\}</math>. If this is not the case, i.e., if <math display="inline">\alpha \leq \beta</math> as multi-indices, then <math display="block"> \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}</math> for each <math>i</math> and the theorem follows. Q.E.D.

ReferencesEdit

Template:Reflist

Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press. Template:Isbn

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