The material derivative

Question

Let $u:S(t) \to \mathbb{R}$ be a scalar field on a surface $S(t)$ parametrised by time. The material derivative is $$Du = u_t + v \cdot \nabla u$$ where $v$ is the velocity. I fail to understand the significance of this.. isn't this just $Du = \frac{du(x,t)}{dt}$ and we apply the chain rule thinking of $x$ to depend on time $t$?

Answer 1

No, the chain rule thinking of $x = x(t)$ would still give you a derivative that is essentially just a derivative with respect to time.

The material derivative gives you something that depends on both time and space -- for instance, the pressure of a fluid might vary with time, but also on the depth of the fluid element in the tank. Even in a fluid that is invariant w/r.t. time, there will be a pressure gradient that exists based on spatial coordinates. The material derivative gives you the total of both of those factors.