# The material derivative

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 $t$?

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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.