= Velocity field
{title2=$\mathbf u(\mathbf x,t)$}
A <velocity field> assigns a <velocity vector> $\mathbf v(\mathbf x,t)$ to each spatial position and <time> in a continuum. Following a material particle means solving $d\mathbf x/dt=\mathbf v(\mathbf x,t)$ with its initial position. The resulting <Lagrangian trajectory> samples the field at changing positions.
The field's spatial variation is the <velocity gradient>, whose symmetric part gives local <strain rate> and whose <trace> gives <velocity divergence>. Even when $\partial_t\mathbf v=0$, a moving particle can have <convective acceleration> $(\mathbf v\cdot\nabla)\mathbf v$. A steady flow narrowing through a pipe demonstrates this: fixed sensors give unchanging readings while each passing parcel speeds up. <Streamlines> are curves tangent to the instantaneous <velocity field>; they coincide with particle paths only under the appropriate steady-flow conditions.
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