Material Derivative

The material derivative is defined along integral curves of the flow. This has some subtle implications. For instance, suppose $f$ represents the temperature of the fluid in a swimming pool, and suppose the pool's temperature decreases as depth increases, and that this is a steady state. Then ${\frac {\partial f}{\partial t}}=0$ everywhere in the pool, but a particle travelling from the deep end to the shallow end will have a nonzero material derivative, because ${\vec {u}}\cdot \nabla f\neq 0$.