= Displacement gradient
{title2=$H=\nabla\mathbf u$}
For displacement $u(X)$ in reference coordinates, the <displacement gradient> is $H_{ij}=\partial u_i/\partial X_j$. It records how nearby points move differently. The <deformation gradient> is $F=I+H$.
In small-strain kinematics, $\epsilon=(H+H^T)/2$ describes <strain>, while the antisymmetric part gives infinitesimal <rotation>. For a rigid translation, $H=0$. For $u=(\alpha X_1,0,0)$, $H_{11}=\alpha$ describes uniform axial extension. For finite <rotations>, the linearized <strain> is not an exact rotation-invariant measure, so a finite-deformation <strain> tensor is needed.
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