This comes from Wiki. I just liked this description for distinction between Eulerian and Lagrangian.
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This static position derivative is called the Eulerian derivative.
An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.
If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature, by swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer’s changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.
That is, the path follows the fluid current described by the fluid’s velocity field v. So, the material derivative of the scalar φ is:
