  We present a unified derivation of covariant time derivatives, which
transform as tensors under a time-dependent coordinate change. Such derivatives
are essential for formulating physical laws in a frame-independent manner.
Three specific derivatives are described: convective, corotational, and
directional. The covariance is made explicit by working in arbitrary
time-dependent coordinates, instead of restricting to Eulerian (fixed) or
Lagrangian (material) coordinates. The commutator of covariant time and space
derivatives is interpreted in terms of a time-curvature that shares many
properties of the Riemann curvature tensor, and reflects nontrivial
time-dependence of the metric.
