Covariant time derivatives for dynamical systems

We present a unified derivation of covariant time derivatives, which transf orm as tensors under a time-dependent coordinate change. Such derivatives a re essential for formulating physical laws in a frame-independent manner. T hree specific derivatives are described: convective, corotational and direc tional. The covariance is made explicit by working in arbitrary time-depend ent coordinates, instead of restricting to Eulerian or Lagrangian coordinat es. The commutator of covariant time and space derivatives is interpreted i n terms of a time-curvature that shares many properties of the Riemann curv ature tensor, and reflects nontrivial time dependence of the metric.