Transport operator along a vector field

For a vector field $b(t,x)$ , the transport operator along $b$ is

T_b=b\cdot\nabla=\sum_{j=1}^d b_j(t,x)\partial_{x_j}.

For a differentiable scalar $q$ , $T_bq$ is its spatial directional derivative along the vector $b$ at that point. For fixed $b$ , it is linear in $q$ .

Time dependence

The material derivative adds the explicit time derivative: $D_t=\partial_t+T_b$ . Even if $b$ depends on time, the operator $T_b$ differentiates only the spatial argument of the field on which it acts.

For Cartesian vector components it acts componentwise. Position-dependent bases must also be differentiated when transporting a vector.