Material derivative - Knowlpedia

For a velocity field $u$ , the material derivative is

D_t=\partial_t+u\cdot\nabla.

It combines explicit time change with the spatial transport operator. For a differentiable scalar field $q$ and a particle trajectory, the chain rule gives

\frac{d}{dt}q(t,X(t))=(D_tq)(t,X(t)).

For Cartesian vector components, the derivative acts componentwise.

Moving bases

When a vector is expressed in a position-dependent basis, differentiating the vector also differentiates the basis. Cylindrical component equations therefore contain curvature terms in addition to applying $\partial_t+u_r\partial_r+(u_\theta/r)\partial_\theta+u_z\partial_z$ to each component.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Section 1.