Multi-index notation generalizes integer index into ordered tuple of indices. It is useful when we have a function with a lot of input variables.

A multi-index is ordered indices of non-negative integers

Definitions

Sum of Components

The degree of a multi-index is defined as

Powers

We define the th power of as

This is the meaning of that monomial term.

The degree of the monomial term equals the degree of the multi-index.

Factorials

Derivatives & Mixed Partials

Use the multi-index to record the number & types of derivatives Given

Define

Example: Higher Derivatives

Which order do we evaluate those partial derivatives? The order in which you compute the partial derivatives doesn’t matter if f has continuous second partial derivatives!