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
Definitions
Sum of Components
The degree of a multi-index
Powers
We define the
This is the meaning of that monomial
The degree of the monomial term equals the degree of the multi-index.
Example: For
Factorials
Example
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!