Multi-index notation
Encyclopedia
The mathematical notation
of multi-indices simplifies formulae used in multivariable calculus
, partial differential equation
s and the theory of distribution
s, by generalising the concept of an integer index
to an ordered tuple
of indices.
of non-negative integers. For multi-indices
and 
one defines:
(or 
), 
, and 
(or 
).
Multinomial theorem
Leibniz formula
For smooth functions f and g
Taylor series
For an analytic function
f in n variables one has
In fact, for a smooth enough function, we have the similar Taylor expansion
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
Integration by parts
For smooth functions with compact support in a bounded domain 
one has
This formula is used for the definition of distribution
s and weak derivative
s.
are multi-indices and 
, then
; if α and β are in {0, 1, 2, . . .}, then
Suppose
, 
, and 
. Then we have that
For each i in {1, . . ., n}, the function
only depends on 
. In the above, each partial differentiation 
therefore reduces to the corresponding ordinary differentiation 
. Hence, from equation (1), it follows that 
vanishes if αi > βi for at least one i in {1, . . ., n}. If this is not the case, i.e., if α ≤ β as multi-indices, then
for each
and the theorem follows. 
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
of multi-indices simplifies formulae used in multivariable calculus
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
, partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s and the theory of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s, by generalising the concept of an integer index
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...
to an ordered tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
of indices.
Multi-index notation
An n-dimensional multi-index is an n-tupleof non-negative integers. For multi-indices
and one defines:- Componentwise sum and difference
-
- Partial order
- Sum of components (absolute value)
- FactorialFactorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
- Factorial
- Binomial coefficientBinomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
- Binomial coefficient
- Multinomial coefficient
where 
- Power
- Higher-order partial derivativePartial derivativeIn 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...
- Higher-order partial derivative
where 
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, (or ), , and (or ).Multinomial theoremMultinomial theoremIn mathematics, the multinomial theorem says 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 to polynomials.-Theorem:...
Multi-binomial theorem
Leibniz formulaLeibniz rule (generalized product rule)In calculus, the general Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by...
For smooth functions f and gTaylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
For an analytic functionAnalytic 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, categories that are similar in some ways, but different in others...
f in n variables one has
In fact, for a smooth enough function, we have the similar Taylor expansion
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
General partial differential operator
A formal N-th order partial differential operator in n variables is written asIntegration by partsIntegration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
For smooth functions with compact support in a bounded domain one hasThis formula is used for the definition of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s and weak derivative
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...
s.
An example theorem
If are multi-indices and , thenProof
The proof follows from the power rule for the ordinary derivativeDifferential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
; if α and β are in {0, 1, 2, . . .}, then
Suppose
, , and . Then we have that For each i in {1, . . ., n}, the function
only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation (1), it follows that vanishes if αi > βi for at least one i in {1, . . ., n}. If this is not the case, i.e., if α ≤ β as multi-indices, thenfor each
and the theorem follows.