Multi-index notation
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 -tupleof non-negative integers.
For multi-indices and, one defines:
;Componentwise sum and difference
;Partial order
;Sum of components
;Factorial
;Binomial coefficient
;Multinomial coefficient
;Power
;Higher-order partial derivative
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, ,, and .;Multinomial theorem
;Multi-binomial theorem
;Leibniz formula
;Taylor series
;General linear partial differential operator
;Integration by parts
An example theorem
If are multi-indices and, thenProof
The proof follows from the power rule for the ordinary derivative; if α and β are in, thenSuppose,, and. Then we have that
For each in, the function only depends on. In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation. Hence, from equation, it follows that vanishes if for at least one in. If this is not the case, i.e., if as multi-indices, then
for each and the theorem follows. Q.E.D.