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. 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. An n-dimensional multi-index is an n-tuple of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted N 0 n {displaystyle mathbb {N} _{0}^{n}} ). For multi-indices α , β ∈ N 0 n {displaystyle alpha ,eta in mathbb {N} _{0}^{n}} and x = ( x 1 , x 2 , … , x n ) ∈ R n {displaystyle x=(x_{1},x_{2},ldots ,x_{n})in mathbb {R} ^{n}} one defines: where k := | α | ∈ N 0 {displaystyle k:=|alpha |in mathbb {N} _{0}} . where ∂ i α i := ∂ α i / ∂ x i α i {displaystyle partial _{i}^{alpha _{i}}:=partial ^{alpha _{i}}/partial x_{i}^{alpha _{i}}} (see also 4-gradient). 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 ∈ C n {displaystyle x,y,hin mathbb {C} ^{n}} (or R n {displaystyle mathbb {R} ^{n}} ), α , ν ∈ N 0 n {displaystyle alpha , u in mathbb {N} _{0}^{n}} , and f , g , a α : C n → C {displaystyle f,g,a_{alpha }colon mathbb {C} ^{n} o mathbb {C} } (or R n → R {displaystyle mathbb {R} ^{n} o mathbb {R} } ). 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...(xn+yn)αn. For smooth functions f and g