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. 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 ). For multi-indices and one defines: Componentwise sum and difference Partial order Sum of components (absolute value) Factorial Binomial coefficient Multinomial coefficient where . Power Higher-order partial derivative where (see also 4-gradient). Sometimes the notation is also used. 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 theorem Multi-binomial theorem 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. 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 General linear partial differential operator A formal linear N-th order partial differential operator in n variables is written as Integration by parts For smooth functions with compact support in a bounded domain one has This formula is used for the definition of distributions and weak derivatives. If are multi-indices and , then The proof follows from the power rule for the ordinary derivative; 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 .