When we consider vector functions of coordinates, we have a double
helping of complexity. First, there are typically *several*
coordinates -
for example - that themselves may form a
vector. Second, the function (physical quantity of interest) may be a
vector, or even a tensor. This means that we can take a vector-like
derivative of a scalar function of vector coordinates and produce a
vector! Alternatively, we can take derivatives that *both* act on
the underlying vector coordinates *and* select out and transform
specific components of the vector quantity itself in specific ways. As
was the case for multiplication of scalars and vectors, we won't have
just one kind - we may end up with three, or four, or more! Indeed,
some of our derivatives will echo the multiplication rules algebraically
specified above.

Robert G. Brown 2017-07-11