Vector Differentiation

When we consider vector functions of coordinates, we have a double helping of complexity. First, there are typically several coordinates - $(x,y,z,t)$ 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.