In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) for a material element subjected to a space-and-time-dependent velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.
For example, in fluid dynamics, take the case that the velocity field under consideration is the flow velocity itself, and the quantity of interest is the temperature of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its pathline (trajectory) while following the fluid flow.
Read more about Material Derivative: Names, Definition, Development, Orthogonal Coordinates
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