= Saint-Venant strain compatibility and differential identities
{c}
{title2=$C_{ij}=\epsilon_{ik\ell}\epsilon_{jmn}e_{\ell n,km}=0,\quad\partial_iC_{ij}=0$}
= Saint-Venant compatibility equations
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{synonym}
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 $\epsilon_{ik\ell}$ 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.
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