Saint-Venant strain compatibility and differential identities ($C_{ij}={ϵ}_{ik\ell }{ϵ}_{jmn}{e}_{\ell n,km}=0,{\partial }_i{C}_{ij}=0$)

For a infinitesimal strain tensor $e=(\nabla u+\nabla u^T)/2$, commuting displacement derivatives gives the symmetric incompatibility tensor $C=0$. Its six component equations are the Saint-Venant compatibility equations. They satisfy three differential identities because ${\partial }_i{\partial }_k$ is symmetric while ${ϵ}_{ik\ell }$ is antisymmetric. Thus there are three independent differential constraints, although keeping all six equations is a coordinate-symmetric way to state them. At a nonzero Fourier wavevector along the third axis the constrained components are $e_{11},e_{22},e_{12}$, while $e_{13},e_{23},e_{33}$ arise from the three displacement components. On a simply connected region compatibility gives local integrability up to a rigid motion; on a multiply connected region circulation conditions must also be imposed.