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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.

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.

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They both give you the exact same expression so are equivalent. $$\dfrac{DF(\mathbf{x}(t),t)}{Dt} = F_t + \mathbf{u}\cdot\nabla F = \dfrac{\text{d} F(\mathbf{x}(t),t)}{\text{d}t}$$ – Angus Leck Apr 30 at 0:16

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