Part of a series of articles about
Calculus
${\displaystyle f(a)-f(b)=\int _{b}^{a}f'(t)\,dt}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis

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 properties

An n-dimensional multi-index is an ${\textstyle n}$-tuple

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

of non-negative integers (i.e. an element of the ${\textstyle n}$-dimensional set of natural numbers, denoted ${\displaystyle \mathbb {N} _{0}^{n}}$).

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

Componentwise sum and difference

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

Partial order

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

Sum of components (absolute value)

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

Factorial

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

Binomial coefficient

${\displaystyle {\binom {\alpha }{\beta }}={\binom {\alpha _{1}}{\beta _{1}}}{\binom {\alpha _{2}}{\beta _{2}}}\cdots {\binom {\alpha _{n}}{\beta _{n}}}={\frac {\alpha$ !}{\beta !(\alpha -\beta )!}}}

Multinomial coefficient

$${\displaystyle {\binom {k}{\alpha }}={\frac {k!}{\alpha _{1}!\alpha _{2}!\cdots \alpha _{n}!}}={\frac {k!}{\alpha$$ !}}}

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

Power

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

Higher-order partial derivative

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

where ${\displaystyle \partial _{i}^{\alpha _{i}}:=\partial ^{\alpha _{i}}/\partial x_{i}^{\alpha _{i}}}$ (see also 4-gradient). Sometimes the notation ${\displaystyle D^{\alpha }=\partial ^{\alpha }}$ is also used.^[1]

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

Multinomial theorem

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

Multi-binomial theorem

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

Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x₁ + y₁)^α₁⋯(x_n + y_n)^α_n.

Leibniz formula

For smooth functions ${\textstyle f}$ and ${\textstyle g}$,

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

Taylor series

For an analytic function ${\textstyle f}$ in ${\textstyle n}$ variables one has

$${\displaystyle f(x+h)=\sum _{\alpha \in \mathbb {N} _{0}^{n}}{{\frac {\partial ^{\alpha }f(x)}{\alpha$$ !}}h^{\alpha }}.}

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

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

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

General linear partial differential operator

A formal linear ${\textstyle N}$-th order partial differential operator in ${\textstyle n}$ variables is written as

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

Integration by parts

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

$${\displaystyle \int _{\Omega }u(\partial ^{\alpha }v)\,dx=(-1)^{|\alpha |}\int _{\Omega }{(\partial ^{\alpha }u)v\,dx}.}$$

This formula is used for the definition of distributions and weak derivatives.

An example theorem

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

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

Proof

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

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

(1)

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

$${\displaystyle {\begin{aligned}\partial ^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}}$$

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

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

for each ${\displaystyle i}$ and the theorem follows. Q.E.D.

See also

• Einstein notation
• Index notation
• Ricci calculus

References

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

This article incorporates material from multi-index derivative of a power on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.