After going over the derivation of the material derivative in Cartesian, cylindrical, and spherical coordinates and seeing all the trouble that we had to go through, it is time to present the material derivative in a vector invariant form.
An invariant vector form is independent of the coordinate system used. The form will be the same for all coordinate. Of course, vector operations (i.e. gradient and curl) are not the same for different coordinate systems [see this post] once they are expanded; but the gradient is always a gradient and the curl is always a curl - only the expansion is different.
I will follow Karamcheti's explanation for obtaining the vector form of the material derivative. We start by considering a fluid particle at R measured from the origin of the coordinate system and time t. Consider also a generic scalar fluid property Q such as the temperature, pressure, or density. The scalar restriction will be removed once we obtain the general form for the material derivative. At point R and time t, the property is defined as Q(R, t). At time (t + Δt), the fluid particle moves a distance Ds and the fluid property changes accordingly to Q(R + V Δt, t + Δt)
The total change in Q from t to (t + Δt) is
If we substitute Eq. 3 into Eq. 2, we get the following
This is the expression we are looking for; Eq. 6 represents the time derivative of a transported fluid property as seen from an Eulerian point of view. This also works when Q is a vector field, call it A