= Velocity gradient
{title2=$\nabla\mathbf u$}
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The <velocity gradient> describes how a <velocity field> varies between nearby points. With components $L_{ij}=\partial v_i/\partial x_j$, nearby material points separated by $\delta\mathbf x$ have relative <velocity> $L\delta\mathbf x$ to first order. It splits into symmetric $D=(L+L^{\mathsf T})/2$, the <rate-of-strain tensor>, and antisymmetric $W=(L-L^{\mathsf T})/2$, describing local spin.
A rigid <rotation> has $D=0$ despite a nonzero <velocity gradient>: neighbouring points move differently without changing their separation lengths. In simple shear $v_x=\dot\gamma y$, both $D_{xy}=\dot\gamma/2$ and spin are nonzero. The <trace> of $L$ is <velocity divergence>, determining local fractional <volume> change. These pieces separate stretching, rotation and dilation, making the <velocity gradient> more informative than a single <strain rate> in a general three-dimensional flow.
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