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Multi-index notation is a
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
that simplifies formulas used in
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
,
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s and the theory of
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
s, by generalising the concept of an integer
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
to an ordered
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of indices.


Definition and basic properties

An ''n''-dimensional multi-index is an ''n''-
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
:\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of
non-negative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s (i.e. an element of the ''n''-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, denoted \mathbb^n_0). For multi-indices \alpha, \beta \in \mathbb^n_0 and x = (x_1, x_2, \ldots, x_n) \in \mathbb^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 In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
:\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\ ;Sum of components (absolute value) :, \alpha , = \alpha_1 + \alpha_2 + \cdots + \alpha_n ;
Factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
:\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n! ;
Binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
:\binom = \binom\binom\cdots\binom = \frac ;
Multinomial coefficient In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer ...
:\binom = \frac = \frac where k:=, \alpha, \in\mathbb_0. ;
Power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
:x^\alpha = x_1^ x_2^ \ldots x_n^. ;Higher-order
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
:\partial^\alpha = \partial_1^ \partial_2^ \ldots \partial_n^ where \partial_i^:=\partial^ / \partial x_i^ (see also 4-gradient). Sometimes the notation D^ = \partial^ is also used.


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\Complex^n (or \R^n), \alpha,\nu\in\N_0^n, and f,g,a_\alpha\colon\Complex^n\to\Complex (or \R^n\to\R). ;
Multinomial theorem In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer ...
: \left( \sum_^n x_i\right)^k = \sum_ \binom \, x^\alpha ;
Multi-binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
: (x+y)^\alpha = \sum_ \binom \, x^\nu y^. Note that, since is a vector and is a multi-index, the expression on the left is short for . ; Leibniz formula :For smooth functions ''f'' and ''g'' \partial^\alpha(fg) = \sum_ \binom \, \partial^f\,\partial^g. ;
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
:For an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
''f'' in ''n'' variables one has f(x+h) = \sum_ . In fact, for a smooth enough function, we have the similar Taylor expansion f(x+h) = \sum_+R_(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_\frac \int_0^1(1-t)^n\partial^\alpha f(x+th) \, dt. ;General linear
partial differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
:A formal linear ''N''-th order partial differential operator in ''n'' variables is written as P(\partial) = \sum_ . ;
Integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
:For smooth functions with compact support in a bounded domain \Omega \subset \R^n one has \int_ u(\partial^v) \, dx = (-1)^ \int_ . This formula is used for the definition of
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
s and
weak derivative In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method ...
s.


An example theorem

If \alpha,\beta\in\mathbb^n_0 are multi-indices and x=(x_1,\ldots, x_n), then \partial^\alpha x^\beta = \begin \frac x^ & \text~ \alpha\le\beta,\\ 0 & \text \end


Proof

The proof follows from the
power rule In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
for the
ordinary derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
; if ''α'' and ''β'' are in , then 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\partial^\alpha x^\beta&= \frac x_1^ \cdots x_n^\\ &= \frac x_1^ \cdots \frac x_n^.\end For each ''i'' in , the function x_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 (), it follows that \partial^\alpha x^\beta vanishes if ''αi'' > ''βi'' for at least one ''i'' in . If this is not the case, i.e., if ''α'' ≤ ''β'' as multi-indices, then \frac x_i^ = \frac x_i^ for each i and the theorem follows. Q.E.D.


See also

* Einstein notation *
Index notation In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to th ...
* Ricci calculus


References

* Saint Raymond, Xavier (1991). ''Elementary Introduction to the Theory of Pseudodifferential Operators''. Chap 1.1 . CRC Press. {{tensors Combinatorics Mathematical notation Articles containing proofs