Multi-index Notation - Some Applications

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 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...(x_n+y_n)α_n.

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 partial differential operator

A formal 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.