= Covector evolution of a material scalar gradient
{title2=$D_t\nabla\theta=-(\nabla\boldsymbol u)^T\nabla\theta$}
Differentiate the <advection equation> componentwise: $D_t(\partial_i\theta)=-(\partial_i u_j)\partial_j\theta$. Along a <Lagrangian trajectory>, the scalar <gradient> therefore evolves by the negative transpose of the <velocity gradient tensor>. If $F$ is the <deformation gradient>, this says $\boldsymbol k(t)=F(t)^{-T}\boldsymbol k(0)$, as expected for a <covector>. Fluid material vectors and scalar gradients transform oppositely under stretching.
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