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.

Definition and basic properties

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:

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 \R^n),, and (or \R^n\to\R). 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 .

An example theorem

If are multi-indices and, then

Proof

The proof follows from the power rule for the ordinary derivative; if α and β are in, then Suppose, , and. Then we have that For each i in, the function only depends on x_i. In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation d/dx_i. Hence, from equation, it follows that vanishes if for at least one i in. If this is not the case, i.e., if as multi-indices, then for each i and the theorem follows. Q.E.D.