The Kronecker delta function is defined by the rules:

Using this we can reduce the dot product to the following tensor contraction, using the Einstein summation convention:

where we sum repeated indices over all of the orthogonal cartesian coordinate indices without having to write an explicit . We will henceforth use this convention almost all the time to streamline the notation of certain kinds of (vector and tensor) algebra.

The Kronecker delta function is obviously useful for representing the dot product in a compact way. We can similarly invent a symbol that incorporates all of the details of the ways the unit vectors multiply in the cross product, next.