Assume that at point r and time t a fluid particle has a property Q. As this particle moves about, this property will change with time (and space). Again, in the Lagrangian description, Q is only a function of time, i.e.
Once Eqs. 8 through 12 are put together, one obtains the full expression for the material derivative of a vector field in spherical coordinates.
In the next post, I will present an invariant vector form for the material derivative so that we don't have to go through all the hassle of using chain rule differentiation to evaluate the material derivative. But it was worth it to see how it works using good old calculus.