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.